file_path
stringlengths 11
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100
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stringclasses 4
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stringclasses 4
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|---|---|---|---|---|---|---|
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.rec_prim
|
[
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n⊢ Primrec fun a => F a (c a)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\n⊢ let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a);\n Primrec fun a => F a (c a)",
"tactic": "intros F"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun a => F a (c a)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n⊢ Primrec fun a => F a (c a)",
"tactic": "let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>\n let a := p.1.1\n let IH := p.1.2\n let n := p.2.1\n let m := p.2.2\n (IH.get? m).bind fun s =>\n (IH.get? m.unpair.1).bind fun s₁ =>\n (IH.get? m.unpair.2).map fun s₂ =>\n cond n.bodd\n (cond n.div2.bodd (rf a (ofNat Code m) s)\n (pc a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂))\n (cond n.div2.bodd (co a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂)\n (pr a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s₁ s₂))"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\n⊢ Primrec fun a => F a (c a)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun a => F a (c a)",
"tactic": "have : Primrec G₁ := by\n refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n unfold Primrec₂\n refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp\n (snd.comp snd))).comp fst) _\n unfold Primrec₂\n refine'\n option_map\n ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <|\n fst.comp fst)\n _\n have a : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.1.1) :=\n fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n have n : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.1) :=\n fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n have m : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.2) :=\n snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n have m₁ := fst.comp (Primrec.unpair.comp m)\n have m₂ := snd.comp (Primrec.unpair.comp m)\n have s : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.2) :=\n snd.comp (fst.comp fst)\n have s₁ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.2) :=\n snd.comp fst\n have s₂ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.2) :=\n snd\n have h₁ := hrf.comp <| a.pair (((Primrec.ofNat Code).comp m).pair s)\n have h₂ := hpc.comp <| a.pair (((Primrec.ofNat Code).comp m₁).pair <|\n ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)\n have h₃ := hco.comp <| a.pair\n (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)\n have h₄ := hpr.comp <| a.pair\n (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)\n unfold Primrec₂\n exact\n (nat_bodd.comp n).cond\n ((nat_bodd.comp <| nat_div2.comp n).cond h₁ h₂)\n (cond (nat_bodd.comp <| nat_div2.comp n) h₃ h₄)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun a => F a (c a)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\n⊢ Primrec fun a => F a (c a)",
"tactic": "let G : α → List σ → Option σ := fun a IH =>\n IH.length.casesOn (some (z a)) fun n =>\n n.casesOn (some (s a)) fun n =>\n n.casesOn (some (l a)) fun n =>\n n.casesOn (some (r a)) fun n => G₁ ((a, IH), n, n.div2.div2)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\n⊢ Primrec fun a => F a (c a)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun a => F a (c a)",
"tactic": "have : Primrec₂ G := by\n unfold Primrec₂\n refine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n unfold Primrec₂\n refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n unfold Primrec₂\n refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n unfold Primrec₂\n refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n unfold Primrec₂\n exact this.comp <|\n ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n snd.pair <| nat_div2.comp <| nat_div2.comp snd"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).fst\n (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).snd =\n some ((fun a n => F a (ofNat Code n)) a n)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\n⊢ Primrec fun a => F a (c a)",
"tactic": "refine'\n ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to₂ fun a n => _).comp\n _root_.Primrec.id <| encode_iff.2 hc).of_eq\n fun a => by simp"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).fst\n (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).snd =\n some ((fun a n => F a (ofNat Code n)) a n)",
"tactic": "simp (config := { zeta := false })"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (List.length\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"tactic": "simp only []"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (List.length\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"tactic": "rw [List.length_map, List.length_range]"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"tactic": "let m := n.div2.div2"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ Nat.rec (some (z a))\n (fun n_1 n_ih =>\n Nat.rec (some (s a))\n (fun n_2 n_ih =>\n Nat.rec (some (l a))\n (fun n_3 n_ih =>\n Nat.rec (some (r a))\n (fun n_4 n_ih =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (div2 (div2 n_4)))\n fun s_1 =>\n Option.bind\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).fst)\n fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n_4 then\n bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n else\n pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n bif bodd (div2 n_4) then\n co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂\n else\n pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n (ofNat Code (unpair (div2 (div2 n_4))).snd) s₁ s₂)\n (List.get?\n (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n (unpair (div2 (div2 n_4))).snd))\n n_3)\n n_2)\n n_1)\n (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n some\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"tactic": "show\n G₁ ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =\n some (F a (ofNat Code (n + 4)))"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"tactic": "have hm : m < n + 4 := by\n simp [Nat.div2_val]\n exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _))"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"tactic": "have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"tactic": "have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (n + 4))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ G₁ ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))",
"tactic": "simp [List.get?_map, List.get?_range, hm, m1, m2]"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNatCode (n + 4))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (n + 4))",
"tactic": "rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]"
},
{
"state_after": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (match bodd n, bodd (div2 n) with\n | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, true => rfind' (ofNatCode (div2 (div2 n))))",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNatCode (n + 4))",
"tactic": "simp [ofNatCode]"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n⊢ (bif bodd n then\n bif bodd (div2 n) then\n rf a (ofNat Code (div2 (div2 n)))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n else\n pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n bif bodd (div2 n) then\n co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n else\n pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n (match bodd n, bodd (div2 n) with\n | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n | true, true => rfind' (ofNatCode (div2 (div2 n))))",
"tactic": "cases n.bodd <;> cases n.div2.bodd <;> rfl"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s =>\n Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.snd.fst then\n bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.snd.fst) then\n co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.snd (unpair p.snd.snd).snd)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec G₁",
"tactic": "refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun p =>\n (fun p s =>\n Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.snd.fst then\n bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.snd.fst) then\n co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.snd (unpair p.snd.snd).snd))\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s =>\n Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.snd.fst then\n bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.snd.fst) then\n co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.snd (unpair p.snd.snd).snd)",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.fst.snd.fst then\n bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n else pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.fst.snd.fst) then\n co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd)",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun p =>\n (fun p s =>\n Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.snd.fst then\n bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.snd.fst) then\n co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂\n else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.snd (unpair p.snd.snd).snd))\n p.fst p.snd",
"tactic": "refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp\n (snd.comp snd))).comp fst) _"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun p =>\n (fun p s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.fst.snd.fst then\n bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n else\n pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.fst.snd.fst) then\n co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd))\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.fst.snd.fst then\n bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n else pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.fst.snd.fst) then\n co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd)",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec fun p =>\n (fun p s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd p.fst.snd.fst then\n bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n else\n pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n bif bodd (div2 p.fst.snd.fst) then\n co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂\n else\n pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s₁ s₂)\n (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd))\n p.fst p.snd",
"tactic": "refine'\n option_map\n ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <|\n fst.comp fst)\n _"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have a : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.1.1) :=\n fst.comp (fst.comp <| fst.comp <| fst.comp fst)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have n : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.1) :=\n fst.comp (snd.comp <| fst.comp <| fst.comp fst)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have m : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.2) :=\n snd.comp (snd.comp <| fst.comp <| fst.comp fst)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have m₁ := fst.comp (Primrec.unpair.comp m)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have m₂ := snd.comp (Primrec.unpair.comp m)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have s : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.2) :=\n snd.comp (fst.comp fst)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have s₁ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.2) :=\n snd.comp fst"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have s₂ : Primrec (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.2) :=\n snd"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have h₁ := hrf.comp <| a.pair (((Primrec.ofNat Code).comp m).pair s)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have h₂ := hpc.comp <| a.pair (((Primrec.ofNat Code).comp m₁).pair <|\n ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have h₃ := hco.comp <| a.pair\n (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₄ :\n Primrec fun a =>\n pr\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "have h₄ := hpr.comp <| a.pair\n (((Primrec.ofNat Code).comp m₁).pair <| ((Primrec.ofNat Code).comp m₂).pair <| s₁.pair s₂)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₄ :\n Primrec fun a =>\n pr\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec fun p =>\n (fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂)\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₄ :\n Primrec fun a =>\n pr\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec₂ fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂",
"tactic": "unfold Primrec₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns✝ : α → σ\nhs : Primrec s✝\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s✝ a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm₁ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm₂ : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns₁ : Primrec fun p => p.fst.snd\ns₂ : Primrec fun p => p.snd\nh₁ :\n Primrec fun a =>\n rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh₂ :\n Primrec fun a =>\n pc\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₃ :\n Primrec fun a =>\n co\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh₄ :\n Primrec fun a =>\n pr\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n⊢ Primrec fun p =>\n (fun p s₂ =>\n bif bodd p.fst.fst.snd.fst then\n bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n else\n pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n bif bodd (div2 p.fst.fst.snd.fst) then\n co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂\n else\n pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n p.snd s₂)\n p.fst p.snd",
"tactic": "exact\n (nat_bodd.comp n).cond\n ((nat_bodd.comp <| nat_div2.comp n).cond h₁ h₂)\n (cond (nat_bodd.comp <| nat_div2.comp n) h₃ h₄)"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p => G p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ G",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n =>\n Nat.casesOn n (some (s p.fst)) fun n =>\n Nat.casesOn n (some (l p.fst)) fun n =>\n Nat.casesOn n (some (r p.fst)) fun n => G₁ ((p.fst, p.snd), n, div2 (div2 n)))\n n",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p => G p.fst p.snd",
"tactic": "refine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n =>\n Nat.casesOn n (some (s p.fst)) fun n =>\n Nat.casesOn n (some (l p.fst)) fun n =>\n Nat.casesOn n (some (r p.fst)) fun n => G₁ ((p.fst, p.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n =>\n Nat.casesOn n (some (s p.fst)) fun n =>\n Nat.casesOn n (some (l p.fst)) fun n =>\n Nat.casesOn n (some (r p.fst)) fun n => G₁ ((p.fst, p.snd), n, div2 (div2 n)))\n n",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n =>\n Nat.casesOn n (some (l p.fst.fst)) fun n =>\n Nat.casesOn n (some (r p.fst.fst)) fun n => G₁ ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n n",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n =>\n Nat.casesOn n (some (s p.fst)) fun n =>\n Nat.casesOn n (some (l p.fst)) fun n =>\n Nat.casesOn n (some (r p.fst)) fun n => G₁ ((p.fst, p.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"tactic": "refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n =>\n Nat.casesOn n (some (l p.fst.fst)) fun n =>\n Nat.casesOn n (some (r p.fst.fst)) fun n => G₁ ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n =>\n Nat.casesOn n (some (l p.fst.fst)) fun n =>\n Nat.casesOn n (some (r p.fst.fst)) fun n => G₁ ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n n",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G₁ ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n))) n",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n =>\n Nat.casesOn n (some (l p.fst.fst)) fun n =>\n Nat.casesOn n (some (r p.fst.fst)) fun n => G₁ ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"tactic": "refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G₁ ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n =>\n (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G₁ ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n))) n",
"tactic": "unfold Primrec₂"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n => (fun n => G₁ ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p =>\n (fun p n =>\n (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G₁ ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n)))\n n)\n p.fst p.snd",
"tactic": "refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p => (fun p n => (fun n => G₁ ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n) p.fst p.snd",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec₂ fun p n => (fun n => G₁ ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n",
"tactic": "unfold Primrec₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\n⊢ Primrec fun p => (fun p n => (fun n => G₁ ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n) p.fst p.snd",
"tactic": "exact this.comp <|\n ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n snd.pair <| nat_div2.comp <| nat_div2.comp snd"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\n⊢ F (id a) (ofNat Code (encode (c a))) = F a (c a)",
"tactic": "simp"
},
{
"state_after": "case succ.succ.succ.zero\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))\n\ncase succ.succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))",
"state_before": "case succ.succ.succ\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =\n some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n)))))",
"tactic": "cases' n with n"
},
{
"state_after": "case succ.succ.succ.zero\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\n⊢ G a [F a zero, F a succ, F a left] = some (F a (ofNatCode (Nat.succ (Nat.succ (Nat.succ 0)))))",
"state_before": "case succ.succ.succ.zero\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\n⊢ G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))",
"tactic": "simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]"
},
{
"state_after": "no goals",
"state_before": "case succ.succ.succ.zero\nα : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\n⊢ G a [F a zero, F a succ, F a left] = some (F a (ofNatCode (Nat.succ (Nat.succ (Nat.succ 0)))))",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ n / 2 / 2 < n + 4",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ m < n + 4",
"tactic": "simp [Nat.div2_val]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nc : α → Code\nhc : Primrec c\nz : α → σ\nhz : Primrec z\ns : α → σ\nhs : Primrec s\nl : α → σ\nhl : Primrec l\nr : α → σ\nhr : Primrec r\npr : α → Code → Code → σ → σ → σ\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : α → Code → Code → σ → σ → σ\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : α → Code → Code → σ → σ → σ\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : α → Code → σ → σ\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG₁ : (α × List σ) × ℕ × ℕ → Option σ :=\n fun p =>\n let a := p.fst.fst;\n let IH := p.fst.snd;\n let n := p.snd.fst;\n let m := p.snd.snd;\n Option.bind (List.get? IH m) fun s =>\n Option.bind (List.get? IH (unpair m).fst) fun s₁ =>\n Option.map\n (fun s₂ =>\n bif bodd n then\n bif bodd (div2 n) then rf a (ofNat Code m) s\n else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else\n bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂\n else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s₁ s₂)\n (List.get? IH (unpair m).snd)\nthis✝ : Primrec G₁\nG : α → List σ → Option σ :=\n fun a IH =>\n Nat.casesOn (List.length IH) (some (z a)) fun n =>\n Nat.casesOn n (some (s a)) fun n =>\n Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, div2 (div2 n))\nthis : Primrec₂ G\na : α\nn : ℕ\nm : ℕ := div2 (div2 n)\n⊢ n / 2 / 2 < n + 4",
"tactic": "exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _))"
}
] |
[
492,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_ne
|
[
{
"state_after": "case a\nα : Type ?u.412614\nβ : Type u_1\nγ : Type ?u.412620\np q : α → Prop\ninst✝³ : DecidablePred p\ninst✝² : DecidablePred q\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset β\nb a✝ : β\n⊢ a✝ ∈ filter (fun a => b ≠ a) s ↔ a✝ ∈ erase s b",
"state_before": "α : Type ?u.412614\nβ : Type u_1\nγ : Type ?u.412620\np q : α → Prop\ninst✝³ : DecidablePred p\ninst✝² : DecidablePred q\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset β\nb : β\n⊢ filter (fun a => b ≠ a) s = erase s b",
"tactic": "ext"
},
{
"state_after": "case a\nα : Type ?u.412614\nβ : Type u_1\nγ : Type ?u.412620\np q : α → Prop\ninst✝³ : DecidablePred p\ninst✝² : DecidablePred q\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset β\nb a✝ : β\n⊢ a✝ ∈ s ∧ ¬b = a✝ ↔ ¬a✝ = b ∧ a✝ ∈ s",
"state_before": "case a\nα : Type ?u.412614\nβ : Type u_1\nγ : Type ?u.412620\np q : α → Prop\ninst✝³ : DecidablePred p\ninst✝² : DecidablePred q\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset β\nb a✝ : β\n⊢ a✝ ∈ filter (fun a => b ≠ a) s ↔ a✝ ∈ erase s b",
"tactic": "simp only [mem_filter, mem_erase, Ne.def, decide_not, Bool.not_eq_true', decide_eq_false_iff_not]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.412614\nβ : Type u_1\nγ : Type ?u.412620\np q : α → Prop\ninst✝³ : DecidablePred p\ninst✝² : DecidablePred q\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\ns : Finset β\nb a✝ : β\n⊢ a✝ ∈ s ∧ ¬b = a✝ ↔ ¬a✝ = b ∧ a✝ ∈ s",
"tactic": "tauto"
}
] |
[
2942,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2938,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_const
|
[] |
[
1048,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1047,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.smul_apply
|
[] |
[
688,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
687,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.isRefl
|
[] |
[
343,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
11
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Finset.mem_memberSubfamily
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ erase a_1 a = s) ↔ insert a s ∈ 𝒜 ∧ ¬a ∈ s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ memberSubfamily a 𝒜 ↔ insert a s ∈ 𝒜 ∧ ¬a ∈ s",
"tactic": "simp_rw [memberSubfamily, mem_image, mem_filter]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ erase a_1 a = s) → insert a s ∈ 𝒜 ∧ ¬a ∈ s",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ erase a_1 a = s) ↔ insert a s ∈ 𝒜 ∧ ¬a ∈ s",
"tactic": "refine' ⟨_, fun h => ⟨insert a s, ⟨h.1, by simp⟩, erase_insert h.2⟩⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\ns : Finset α\nhs1 : s ∈ 𝒜\nhs2 : a ∈ s\n⊢ insert a (erase s a) ∈ 𝒜 ∧ ¬a ∈ erase s a",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ (∃ a_1, (a_1 ∈ 𝒜 ∧ a ∈ a_1) ∧ erase a_1 a = s) → insert a s ∈ 𝒜 ∧ ¬a ∈ s",
"tactic": "rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\ns : Finset α\nhs1 : s ∈ 𝒜\nhs2 : a ∈ s\n⊢ s ∈ 𝒜 ∧ ¬a ∈ erase s a",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\ns : Finset α\nhs1 : s ∈ 𝒜\nhs2 : a ∈ s\n⊢ insert a (erase s a) ∈ 𝒜 ∧ ¬a ∈ erase s a",
"tactic": "rw [insert_erase hs2]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\na : α\ns : Finset α\nhs1 : s ∈ 𝒜\nhs2 : a ∈ s\n⊢ s ∈ 𝒜 ∧ ¬a ∈ erase s a",
"tactic": "exact ⟨hs1, not_mem_erase _ _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝒜 ∧ ¬a ∈ s\n⊢ a ∈ insert a s",
"tactic": "simp"
}
] |
[
69,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.iSup_sInfGen_nonempty
|
[
{
"state_after": "case inl\nα : Type u_1\nm : Set (OuterMeasure α)\nh : Set.Nonempty m\n⊢ (⨆ (_ : Set.Nonempty ∅), sInfGen m ∅) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅\n\ncase inr\nα : Type u_1\nm : Set (OuterMeasure α)\nh : Set.Nonempty m\nt : Set α\nht : Set.Nonempty t\n⊢ (⨆ (_ : Set.Nonempty t), sInfGen m t) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ t",
"state_before": "α : Type u_1\nm : Set (OuterMeasure α)\nh : Set.Nonempty m\nt : Set α\n⊢ (⨆ (_ : Set.Nonempty t), sInfGen m t) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ t",
"tactic": "rcases t.eq_empty_or_nonempty with (rfl | ht)"
},
{
"state_after": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨆ (_ : Set.Nonempty ∅), sInfGen m ∅) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅",
"state_before": "case inl\nα : Type u_1\nm : Set (OuterMeasure α)\nh : Set.Nonempty m\n⊢ (⨆ (_ : Set.Nonempty ∅), sInfGen m ∅) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅",
"tactic": "rcases h with ⟨μ, hμ⟩"
},
{
"state_after": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅) = ⊥",
"state_before": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨆ (_ : Set.Nonempty ∅), sInfGen m ∅) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅",
"tactic": "rw [eq_false Set.not_nonempty_empty, iSup_false, eq_comm]"
},
{
"state_after": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), 0) = ⊥",
"state_before": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ ∅) = ⊥",
"tactic": "simp_rw [empty']"
},
{
"state_after": "case inl.intro.h\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), 0) ≤ ⊥",
"state_before": "case inl.intro\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), 0) = ⊥",
"tactic": "apply bot_unique"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.h\nα : Type u_1\nm : Set (OuterMeasure α)\nμ : OuterMeasure α\nhμ : μ ∈ m\n⊢ (⨅ (μ : OuterMeasure α) (_ : μ ∈ m), 0) ≤ ⊥",
"tactic": "refine' iInf_le_of_le μ (iInf_le _ hμ)"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nm : Set (OuterMeasure α)\nh : Set.Nonempty m\nt : Set α\nht : Set.Nonempty t\n⊢ (⨆ (_ : Set.Nonempty t), sInfGen m t) = ⨅ (μ : OuterMeasure α) (_ : μ ∈ m), ↑μ t",
"tactic": "simp [ht, sInfGen_def]"
}
] |
[
1166,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1158,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
ContinuousWithinAt.const_cpow
|
[] |
[
134,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
8
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean
|
CategoryTheory.GrothendieckTopology.sheafify_hom_ext
|
[
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ toPlus J (plusObj J P) ≫ η = toPlus J (plusObj J P) ≫ γ",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ η = γ",
"tactic": "apply J.plus_hom_ext _ _ hQ"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ toPlus J P ≫ toPlus J (plusObj J P) ≫ η = toPlus J P ≫ toPlus J (plusObj J P) ≫ γ",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ toPlus J (plusObj J P) ≫ η = toPlus J (plusObj J P) ≫ γ",
"tactic": "apply J.plus_hom_ext _ _ hQ"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ (toPlus J P ≫ plusMap J (toPlus J P)) ≫ η = (toPlus J P ≫ plusMap J (toPlus J P)) ≫ γ",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ toPlus J P ≫ toPlus J (plusObj J P) ≫ η = toPlus J P ≫ toPlus J (plusObj J P) ≫ γ",
"tactic": "rw [← Category.assoc, ← Category.assoc, ← plusMap_toPlus]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\nP Q : Cᵒᵖ ⥤ D\nη γ : sheafify J P ⟶ Q\nhQ : Presheaf.IsSheaf J Q\nh : toSheafify J P ≫ η = toSheafify J P ≫ γ\n⊢ (toPlus J P ≫ plusMap J (toPlus J P)) ≫ η = (toPlus J P ≫ plusMap J (toPlus J P)) ≫ γ",
"tactic": "exact h"
}
] |
[
589,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat.mul
|
[
{
"state_after": "no goals",
"state_before": "p a b : ℕ\nhp : Fact (Nat.Prime p)\n⊢ a ≠ 0 → b ≠ 0 → padicValNat p (a * b) = padicValNat p a + padicValNat p b",
"tactic": "exact_mod_cast @padicValRat.mul p _ a b"
}
] |
[
408,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
11
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.pow_lt_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.96829\nβ : Type ?u.96832\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ a < ⊤ → ∀ (n : ℕ), a ^ n < ⊤",
"tactic": "simpa only [lt_top_iff_ne_top] using pow_ne_top"
}
] |
[
639,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.Results.val_unique
|
[] |
[
533,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.eq_empty_of_ssubset_singleton
|
[] |
[
802,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.add_hom
|
[] |
[
408,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ioc.coe_one
|
[] |
[
258,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean
|
Matrix.GeneralLinearGroup.ext_iff
|
[] |
[
100,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.liftRel_nil
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct nil) (destruct nil)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\n⊢ LiftRel R nil nil",
"tactic": "rw [liftRel_destruct_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct nil) (destruct nil)",
"tactic": "simp [-liftRel_pure_left, -liftRel_pure_right]"
}
] |
[
1095,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/Data/Rat/Floor.lean
|
Rat.num_lt_succ_floor_mul_den
|
[
{
"state_after": "q : ℚ\n⊢ ↑q.num < (↑⌊q⌋ + 1) * ↑q.den",
"state_before": "q : ℚ\n⊢ q.num < (⌊q⌋ + 1) * ↑q.den",
"tactic": "suffices (q.num : ℚ) < (⌊q⌋ + 1) * q.den by exact_mod_cast this"
},
{
"state_after": "q : ℚ\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"state_before": "q : ℚ\n⊢ ↑q.num < (↑⌊q⌋ + 1) * ↑q.den",
"tactic": "suffices (q.num : ℚ) < (q - fract q + 1) * q.den by\n have : (⌊q⌋ : ℚ) = q - fract q := eq_sub_of_add_eq <| floor_add_fract q\n rwa [this]"
},
{
"state_after": "q : ℚ\n⊢ ↑q.num < ↑q.num + (1 - fract q) * ↑q.den",
"state_before": "q : ℚ\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"tactic": "suffices (q.num : ℚ) < q.num + (1 - fract q) * q.den by\n have : (q - fract q + 1) * q.den = q.num + (1 - fract q) * q.den\n calc\n (q - fract q + 1) * q.den = (q + (1 - fract q)) * q.den := by ring\n _ = q * q.den + (1 - fract q) * q.den := by rw [add_mul]\n _ = q.num + (1 - fract q) * q.den := by simp\n rwa [this]"
},
{
"state_after": "q : ℚ\n⊢ 0 < (1 - fract q) * ↑q.den",
"state_before": "q : ℚ\n⊢ ↑q.num < ↑q.num + (1 - fract q) * ↑q.den",
"tactic": "suffices 0 < (1 - fract q) * q.den by\n rw [← sub_lt_iff_lt_add']\n simpa"
},
{
"state_after": "q : ℚ\nthis : 0 < 1 - fract q\n⊢ 0 < (1 - fract q) * ↑q.den",
"state_before": "q : ℚ\n⊢ 0 < (1 - fract q) * ↑q.den",
"tactic": "have : 0 < 1 - fract q := by\n have : fract q < 1 := fract_lt_one q\n have : 0 + fract q < 1 := by simp [this]\n rwa [lt_sub_iff_add_lt]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : 0 < 1 - fract q\n⊢ 0 < (1 - fract q) * ↑q.den",
"tactic": "exact mul_pos this (by exact_mod_cast q.pos)"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : ↑q.num < (↑⌊q⌋ + 1) * ↑q.den\n⊢ q.num < (⌊q⌋ + 1) * ↑q.den",
"tactic": "exact_mod_cast this"
},
{
"state_after": "q : ℚ\nthis✝ : ↑q.num < (q - fract q + 1) * ↑q.den\nthis : ↑⌊q⌋ = q - fract q\n⊢ ↑q.num < (↑⌊q⌋ + 1) * ↑q.den",
"state_before": "q : ℚ\nthis : ↑q.num < (q - fract q + 1) * ↑q.den\n⊢ ↑q.num < (↑⌊q⌋ + 1) * ↑q.den",
"tactic": "have : (⌊q⌋ : ℚ) = q - fract q := eq_sub_of_add_eq <| floor_add_fract q"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis✝ : ↑q.num < (q - fract q + 1) * ↑q.den\nthis : ↑⌊q⌋ = q - fract q\n⊢ ↑q.num < (↑⌊q⌋ + 1) * ↑q.den",
"tactic": "rwa [this]"
},
{
"state_after": "case this\nq : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n\nq : ℚ\nthis✝ : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\nthis : (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"state_before": "q : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"tactic": "have : (q - fract q + 1) * q.den = q.num + (1 - fract q) * q.den"
},
{
"state_after": "q : ℚ\nthis✝ : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\nthis : (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"state_before": "case this\nq : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n\nq : ℚ\nthis✝ : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\nthis : (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"tactic": "calc\n (q - fract q + 1) * q.den = (q + (1 - fract q)) * q.den := by ring\n _ = q * q.den + (1 - fract q) * q.den := by rw [add_mul]\n _ = q.num + (1 - fract q) * q.den := by simp"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis✝ : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\nthis : (q - fract q + 1) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den\n⊢ ↑q.num < (q - fract q + 1) * ↑q.den",
"tactic": "rwa [this]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ (q - fract q + 1) * ↑q.den = (q + (1 - fract q)) * ↑q.den",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ (q + (1 - fract q)) * ↑q.den = q * ↑q.den + (1 - fract q) * ↑q.den",
"tactic": "rw [add_mul]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : ↑q.num < ↑q.num + (1 - fract q) * ↑q.den\n⊢ q * ↑q.den + (1 - fract q) * ↑q.den = ↑q.num + (1 - fract q) * ↑q.den",
"tactic": "simp"
},
{
"state_after": "q : ℚ\nthis : 0 < (1 - fract q) * ↑q.den\n⊢ ↑q.num - ↑q.num < (1 - fract q) * ↑q.den",
"state_before": "q : ℚ\nthis : 0 < (1 - fract q) * ↑q.den\n⊢ ↑q.num < ↑q.num + (1 - fract q) * ↑q.den",
"tactic": "rw [← sub_lt_iff_lt_add']"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : 0 < (1 - fract q) * ↑q.den\n⊢ ↑q.num - ↑q.num < (1 - fract q) * ↑q.den",
"tactic": "simpa"
},
{
"state_after": "q : ℚ\nthis : fract q < 1\n⊢ 0 < 1 - fract q",
"state_before": "q : ℚ\n⊢ 0 < 1 - fract q",
"tactic": "have : fract q < 1 := fract_lt_one q"
},
{
"state_after": "q : ℚ\nthis✝ : fract q < 1\nthis : 0 + fract q < 1\n⊢ 0 < 1 - fract q",
"state_before": "q : ℚ\nthis : fract q < 1\n⊢ 0 < 1 - fract q",
"tactic": "have : 0 + fract q < 1 := by simp [this]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis✝ : fract q < 1\nthis : 0 + fract q < 1\n⊢ 0 < 1 - fract q",
"tactic": "rwa [lt_sub_iff_add_lt]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : fract q < 1\n⊢ 0 + fract q < 1",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "q : ℚ\nthis : 0 < 1 - fract q\n⊢ 0 < ↑q.den",
"tactic": "exact_mod_cast q.pos"
}
] |
[
131,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lxor'_comm
|
[] |
[
185,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.zpow_add
|
[
{
"state_after": "case intro\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ ↑x ^ (m + n) = ↑x ^ m * ↑x ^ n",
"state_before": "α : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx : ℝ≥0∞\nhx : x ≠ 0\nh'x : x ≠ ⊤\nm n : ℤ\n⊢ x ^ (m + n) = x ^ m * x ^ n",
"tactic": "lift x to ℝ≥0 using h'x"
},
{
"state_after": "case hx\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ x ≠ 0\n\ncase intro\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : x ≠ 0\n⊢ ↑x ^ (m + n) = ↑x ^ m * ↑x ^ n",
"state_before": "case intro\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ ↑x ^ (m + n) = ↑x ^ m * ↑x ^ n",
"tactic": "replace hx : x ≠ 0"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : x ≠ 0\n⊢ ↑x ^ (m + n) = ↑x ^ m * ↑x ^ n",
"tactic": "simp only [← coe_zpow hx, zpow_add₀ hx, coe_mul]"
},
{
"state_after": "no goals",
"state_before": "case hx\nα : Type ?u.764582\nβ : Type ?u.764585\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nm n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ x ≠ 0",
"tactic": "simpa only [Ne.def, coe_eq_zero] using hx"
}
] |
[
1938,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1934,
11
] |
Mathlib/Order/UpperLower/Basic.lean
|
IsUpperSet.inter
|
[] |
[
114,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Order/Lattice.lean
|
le_sup_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\n⊢ a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c",
"tactic": "exact ⟨fun h =>\n (le_total c b).imp\n (fun bc => by rwa [sup_eq_left.2 bc] at h)\n (fun bc => by rwa [sup_eq_right.2 bc] at h),\n fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\nh : a ≤ b ⊔ c\nbc : c ≤ b\n⊢ a ≤ b",
"tactic": "rwa [sup_eq_left.2 bc] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : LinearOrder α\na b c d : α\nh : a ≤ b ⊔ c\nbc : b ≤ c\n⊢ a ≤ c",
"tactic": "rwa [sup_eq_right.2 bc] at h"
}
] |
[
849,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
844,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
AddSubsemigroup.toSubsemigroup'_closure
|
[] |
[
119,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.centroid_eq_centroid_image_of_inj_on
|
[
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "let f : p '' ↑s → ι := fun x => x.property.choose"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "have hf : ∀ x, f x ∈ s ∧ p (f x) = x := fun x => x.property.choose_spec"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "let f' : ps → ι := fun x => f ⟨x, hps ▸ x.property⟩"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "have hf' : ∀ x, f' x ∈ s ∧ p (f' x) = x := fun x => hf ⟨x, hps ▸ x.property⟩"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "have hf'i : Function.Injective f' := by\n intro x y h\n rw [Subtype.ext_iff, ← (hf' x).2, ← (hf' y).2, h]"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "let f'e : ps ↪ ι := ⟨f', hf'i⟩"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\n⊢ centroid k univ (p ∘ ↑f'e) = centroid k univ fun x => ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\n⊢ centroid k s p = centroid k univ fun x => ↑x",
"tactic": "rw [← hu, centroid_map]"
},
{
"state_after": "case e_p.h\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\nx : ↑ps\n⊢ (p ∘ ↑f'e) x = ↑x",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\n⊢ centroid k univ (p ∘ ↑f'e) = centroid k univ fun x => ↑x",
"tactic": "congr with x"
},
{
"state_after": "case e_p.h\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\nx : ↑ps\n⊢ p (f' x) = ↑x",
"state_before": "case e_p.h\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\nx : ↑ps\n⊢ (p ∘ ↑f'e) x = ↑x",
"tactic": "change p (f' x) = ↑x"
},
{
"state_after": "no goals",
"state_before": "case e_p.h\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nhu : map f'e univ = s\nx : ↑ps\n⊢ p (f' x) = ↑x",
"tactic": "rw [(hf' x).2]"
},
{
"state_after": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nx y : ↑ps\nh : f' x = f' y\n⊢ x = y",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\n⊢ Function.Injective f'",
"tactic": "intro x y h"
},
{
"state_after": "no goals",
"state_before": "k : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nx y : ↑ps\nh : f' x = f' y\n⊢ x = y",
"tactic": "rw [Subtype.ext_iff, ← (hf' x).2, ← (hf' y).2, h]"
},
{
"state_after": "case a\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ x ∈ map f'e univ ↔ x ∈ s",
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"tactic": "ext x"
},
{
"state_after": "case a\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ (∃ a, a ∈ univ ∧ ↑f'e a = x) ↔ x ∈ s",
"state_before": "case a\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ x ∈ map f'e univ ↔ x ∈ s",
"tactic": "rw [mem_map]"
},
{
"state_after": "case a.mp\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ (∃ a, a ∈ univ ∧ ↑f'e a = x) → x ∈ s\n\ncase a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ x ∈ s → ∃ a, a ∈ univ ∧ ↑f'e a = x",
"state_before": "case a\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ (∃ a, a ∈ univ ∧ ↑f'e a = x) ↔ x ∈ s",
"tactic": "constructor"
},
{
"state_after": "case a.mp.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\ni : ↑ps\nleft✝ : i ∈ univ\n⊢ ↑f'e i ∈ s",
"state_before": "case a.mp\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ (∃ a, a ∈ univ ∧ ↑f'e a = x) → x ∈ s",
"tactic": "rintro ⟨i, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.mp.intro.intro\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\ni : ↑ps\nleft✝ : i ∈ univ\n⊢ ↑f'e i ∈ s",
"tactic": "exact (hf' i).1"
},
{
"state_after": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ ∃ a, a ∈ univ ∧ ↑f'e a = x",
"state_before": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\n⊢ x ∈ s → ∃ a, a ∈ univ ∧ ↑f'e a = x",
"tactic": "intro hx"
},
{
"state_after": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ ↑f'e { val := p x, property := (_ : p x ∈ ps) } = x",
"state_before": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ ∃ a, a ∈ univ ∧ ↑f'e a = x",
"tactic": "use ⟨p x, hps.symm ▸ Set.mem_image_of_mem _ hx⟩, mem_univ _"
},
{
"state_after": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ p (f' { val := p x, property := (_ : p x ∈ ps) }) = p x",
"state_before": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ ↑f'e { val := p x, property := (_ : p x ∈ ps) } = x",
"tactic": "refine' hi _ (hf' _).1 _ hx _"
},
{
"state_after": "no goals",
"state_before": "case a.mpr\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.521615\ns₂ : Finset ι₂\np : ι → P\nhi : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → p i = p j → i = j\nps : Set P\ninst✝ : Fintype ↑ps\nhps : ps = p '' ↑s\nf : ↑(p '' ↑s) → ι := fun x => Exists.choose (_ : ↑x ∈ p '' ↑s)\nhf : ∀ (x : ↑(p '' ↑s)), f x ∈ s ∧ p (f x) = ↑x\nf' : ↑ps → ι := fun x => f { val := ↑x, property := (_ : ↑x ∈ p '' ↑s) }\nhf' : ∀ (x : ↑ps), f' x ∈ s ∧ p (f' x) = ↑x\nhf'i : Function.Injective f'\nf'e : ↑ps ↪ ι := { toFun := f', inj' := hf'i }\nx : ι\nhx : x ∈ s\n⊢ p (f' { val := p x, property := (_ : p x ∈ ps) }) = p x",
"tactic": "rw [(hf' _).2]"
}
] |
[
974,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
950,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.count_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.353095\nγ : Type ?u.353098\ninst✝ : DecidableEq α\na b : α\n⊢ count a {b} = if a = b then 1 else 0",
"tactic": "simp only [count_cons, ← cons_zero, count_zero, zero_add]"
}
] |
[
2384,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2383,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
nnnorm_inner_le_nnnorm
|
[] |
[
1102,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1101,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.image_sub_const_uIcc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na b c d : α\n⊢ (fun x => x - a) '' [[b, c]] = [[b - a, c - a]]",
"tactic": "simp [sub_eq_add_neg, add_comm]"
}
] |
[
471,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/RingTheory/WittVector/IsPoly.lean
|
WittVector.IsPoly₂.ext
|
[
{
"state_after": "case mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhg : IsPoly₂ p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nhf : IsPoly₂ p f\nhg : IsPoly₂ p g\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y",
"tactic": "obtain ⟨φ, hf⟩ := hf"
},
{
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"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\n⊢ ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y",
"tactic": "intros"
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{
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"state_before": "case mk'.intro.mk'.intro\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\n⊢ f x✝ y✝ = g x✝ y✝",
"tactic": "ext n"
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{
"state_after": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn : ℕ\n⊢ ∀ (n : ℕ), ↑(bind₁ φ) (wittPolynomial p ℤ n) = ↑(bind₁ ψ) (wittPolynomial p ℤ n)",
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"tactic": "rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq' p φ ψ]"
},
{
"state_after": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\n⊢ ↑(bind₁ φ) (wittPolynomial p ℤ k) = ↑(bind₁ ψ) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn : ℕ\n⊢ ∀ (n : ℕ), ↑(bind₁ φ) (wittPolynomial p ℤ n) = ↑(bind₁ ψ) (wittPolynomial p ℤ n)",
"tactic": "intro k"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\n⊢ ∀ (x : Fin 2 × ℕ → ℤ),\n ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\n⊢ ↑(bind₁ φ) (wittPolynomial p ℤ k) = ↑(bind₁ ψ) (wittPolynomial p ℤ k)",
"tactic": "apply MvPolynomial.funext"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\n⊢ ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\n⊢ ∀ (x : Fin 2 × ℕ → ℤ),\n ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"tactic": "intro x"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\n⊢ ↑(MvPolynomial.eval x) (↑(bind₁ φ) (wittPolynomial p ℤ k)) =\n ↑(MvPolynomial.eval x) (↑(bind₁ ψ) (wittPolynomial p ℤ k))",
"tactic": "simp only [hom_bind₁]"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(ghostComponent k) (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })) =\n ↑(ghostComponent k) (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) }))\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nh : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), ↑(ghostComponent n) (f x y) = ↑(ghostComponent n) (g x y)\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "specialize h (ULift ℤ) (mk p fun i => ⟨x (0, i)⟩) (mk p fun i => ⟨x (1, i)⟩) k"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(ghostComponent k) (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })) =\n ↑(ghostComponent k) (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) }))\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i))\n (wittPolynomial p ℤ k)) =\n ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))",
"state_before": "case mk'.intro.mk'.intro.h.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i)) (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k)",
"tactic": "apply (ULift.ringEquiv.symm : ℤ ≃+* _).injective"
},
{
"state_after": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (φ i))\n (wittPolynomial p ℤ k)) =\n ↑(RingEquiv.symm ULift.ringEquiv)\n (↑(eval₂Hom (RingHom.comp (MvPolynomial.eval x) C) fun i => ↑(MvPolynomial.eval x) (ψ i)) (wittPolynomial p ℤ k))",
"tactic": "simp only [← RingEquiv.coe_toRingHom, map_eval₂Hom]"
},
{
"state_after": "case h.e'_2\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n\ncase h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)",
"state_before": "case mk'.intro.mk'.intro.h.h.a\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k)",
"tactic": "convert h using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (φ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n\ncase h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)",
"tactic": "all_goals\n simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]\n apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl\n ext1\n apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl\n ext ⟨b, _⟩\n fin_cases b <;> simp only [coeff_mk, uncurry] <;> rfl"
},
{
"state_after": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (eval₂Hom (RingHom.id ℤ) x) C)) fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i))\n (wittPolynomial p ℤ k) =\n ↑(aeval fun n =>\n peval (ψ n) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff])\n (wittPolynomial p ℤ k)",
"state_before": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (MvPolynomial.eval x) C)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (↑(MvPolynomial.eval x) (ψ i)))\n (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)",
"tactic": "simp only [hf, hg, MvPolynomial.eval, map_eval₂Hom]"
},
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ (fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i)) =\n fun n => peval (ψ n) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"state_before": "case h.e'_3\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.comp (eval₂Hom (RingHom.id ℤ) x) C)) fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i))\n (wittPolynomial p ℤ k) =\n ↑(aeval fun n =>\n peval (ψ n) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff])\n (wittPolynomial p ℤ k)",
"tactic": "apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl"
},
{
"state_after": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ x✝) =\n peval (ψ x✝) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\n⊢ (fun i =>\n ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ i)) =\n fun n => peval (ψ n) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"tactic": "ext1"
},
{
"state_after": "p : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) =\n uncurry ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"state_before": "case h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ ↑(eval₂Hom (RingHom.comp (↑(RingEquiv.symm ULift.ringEquiv)) (RingHom.id ℤ)) fun i =>\n ↑↑(RingEquiv.symm ULift.ringEquiv) (x i))\n (ψ x✝) =\n peval (ψ x✝) ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"tactic": "apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl"
},
{
"state_after": "case h.mk.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\nb : Fin 2\nsnd✝ : ℕ\n⊢ (↑↑(RingEquiv.symm ULift.ringEquiv) (x (b, snd✝))).down =\n (uncurry ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff] (b, snd✝)).down",
"state_before": "p : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\n⊢ (fun i => ↑↑(RingEquiv.symm ULift.ringEquiv) (x i)) =\n uncurry ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff]",
"tactic": "ext ⟨b, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.h\np : ℕ\nR S : Type u\nσ : Type ?u.744260\nidx : Type ?u.744263\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Fact (Nat.Prime p)\nf g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R\nφ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]\nψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ\nhg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]\nR✝ : Type u\n_Rcr✝ : CommRing R✝\nx✝¹ y✝ : 𝕎 R✝\nn k : ℕ\nx : Fin 2 × ℕ → ℤ\nh :\n ↑(aeval (f (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k) =\n ↑(aeval (g (mk p fun i => { down := x (0, i) }) (mk p fun i => { down := x (1, i) })).coeff) (wittPolynomial p ℤ k)\nx✝ : ℕ\nb : Fin 2\nsnd✝ : ℕ\n⊢ (↑↑(RingEquiv.symm ULift.ringEquiv) (x (b, snd✝))).down =\n (uncurry ![(mk p fun i => { down := x (0, i) }).coeff, (mk p fun i => { down := x (1, i) }).coeff] (b, snd✝)).down",
"tactic": "fin_cases b <;> simp only [coeff_mk, uncurry] <;> rfl"
}
] |
[
614,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.union_distrib_right
|
[] |
[
1066,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1065,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean
|
CategoryTheory.Limits.zeroProdIso_inv_snd
|
[
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ (limit.isoLimitCone { cone := BinaryFan.mk 0 (𝟙 X), isLimit := binaryFanZeroLeftIsLimit X }).inv ≫ prod.snd = 𝟙 X",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ (zeroProdIso X).inv ≫ prod.snd = 𝟙 X",
"tactic": "dsimp [zeroProdIso, binaryFanZeroLeft]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\n⊢ (limit.isoLimitCone { cone := BinaryFan.mk 0 (𝟙 X), isLimit := binaryFanZeroLeftIsLimit X }).inv ≫ prod.snd = 𝟙 X",
"tactic": "simp"
}
] |
[
63,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
Filter.Tendsto.uniformity_trans
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.47087\ninst✝ : UniformSpace α\nl : Filter β\nf₁ f₂ f₃ : β → α\nh₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α)\nh₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ (fun x => (f₁ x, f₃ x)) ⁻¹' (s ○ s) ∈ l",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.47087\ninst✝ : UniformSpace α\nl : Filter β\nf₁ f₂ f₃ : β → α\nh₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α)\nh₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)\n⊢ Tendsto (fun x => (f₁ x, f₃ x)) l (𝓤 α)",
"tactic": "refine' le_trans (le_lift'.2 fun s hs => mem_map.2 _) comp_le_uniformity"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.47087\ninst✝ : UniformSpace α\nl : Filter β\nf₁ f₂ f₃ : β → α\nh₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α)\nh₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)\ns : Set (α × α)\nhs : s ∈ 𝓤 α\n⊢ (fun x => (f₁ x, f₃ x)) ⁻¹' (s ○ s) ∈ l",
"tactic": "filter_upwards [mem_map.1 (h₁₂ hs), mem_map.1 (h₂₃ hs)] with x hx₁₂ hx₂₃ using ⟨_, hx₁₂, hx₂₃⟩"
}
] |
[
490,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Algebra/Group/Units.lean
|
val_div_eq_divp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na b c : α\nu₁ u₂ : αˣ\n⊢ ↑(u₁ / u₂) = ↑u₁ /ₚ u₂",
"tactic": "rw [divp, division_def, Units.val_mul]"
}
] |
[
526,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Control/Fold.lean
|
Traversable.foldMap_hom
|
[] |
[
290,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mem_inv_smul_finset_iff₀
|
[] |
[
2044,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2043,
1
] |
Mathlib/Order/Basic.lean
|
gt_iff_lt
|
[] |
[
350,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.zero_eq_coe
|
[] |
[
344,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
20
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
|
le_of_forall_one_lt_le_mul
|
[
{
"state_after": "case intro\nα : Type u\ninst✝⁵ : LinearOrder α\ninst✝⁴ : DenselyOrdered α\ninst✝³ : Monoid α\ninst✝² : ExistsMulOfLE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\nh : ∀ (ε : α), 1 < ε → a ≤ b * ε\nε : α\nhxb : b < b * ε\n⊢ a ≤ b * ε",
"state_before": "α : Type u\ninst✝⁵ : LinearOrder α\ninst✝⁴ : DenselyOrdered α\ninst✝³ : Monoid α\ninst✝² : ExistsMulOfLE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\nh : ∀ (ε : α), 1 < ε → a ≤ b * ε\nx : α\nhxb : b < x\n⊢ a ≤ x",
"tactic": "obtain ⟨ε, rfl⟩ := exists_mul_of_le hxb.le"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\ninst✝⁵ : LinearOrder α\ninst✝⁴ : DenselyOrdered α\ninst✝³ : Monoid α\ninst✝² : ExistsMulOfLE α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\nh : ∀ (ε : α), 1 < ε → a ≤ b * ε\nε : α\nhxb : b < b * ε\n⊢ a ≤ b * ε",
"tactic": "exact h _ ((lt_mul_iff_one_lt_right' b).1 hxb)"
}
] |
[
77,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.elementarilyEquivalent_iff
|
[
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type ?u.269466\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\n⊢ M ≅[L] N ↔ ∀ (φ : Sentence L), M ⊨ φ ↔ N ⊨ φ",
"tactic": "simp only [ElementarilyEquivalent, Set.ext_iff, completeTheory, Set.mem_setOf_eq]"
}
] |
[
797,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
GeneralizedContinuedFraction.IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some
|
[
{
"state_after": "case intro.intro\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq : Stream'.Seq.get? (of v).s n = some gp_n\nifp : IntFractPair K\nstream_succ_nth_eq : IntFractPair.stream v (n + 1) = some ifp\ngp_n_eq : { a := 1, b := ↑ifp.b } = gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ ↑ifp.b = gp_n.b",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq : Stream'.Seq.get? (of v).s n = some gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ ↑ifp.b = gp_n.b",
"tactic": "obtain ⟨ifp, stream_succ_nth_eq, gp_n_eq⟩ :\n ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ Pair.mk 1 (ifp.b : K) = gp_n := by\n unfold of IntFractPair.seq1 at s_nth_eq\n simpa [Stream'.Seq.get?_tail, Stream'.Seq.map_get?] using s_nth_eq"
},
{
"state_after": "case intro.intro.refl\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nifp : IntFractPair K\nstream_succ_nth_eq : IntFractPair.stream v (n + 1) = some ifp\ns_nth_eq : Stream'.Seq.get? (of v).s n = some { a := 1, b := ↑ifp.b }\n⊢ ∃ ifp_1, IntFractPair.stream v (n + 1) = some ifp_1 ∧ ↑ifp_1.b = { a := 1, b := ↑ifp.b }.b",
"state_before": "case intro.intro\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq : Stream'.Seq.get? (of v).s n = some gp_n\nifp : IntFractPair K\nstream_succ_nth_eq : IntFractPair.stream v (n + 1) = some ifp\ngp_n_eq : { a := 1, b := ↑ifp.b } = gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ ↑ifp.b = gp_n.b",
"tactic": "cases gp_n_eq"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refl\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\nifp : IntFractPair K\nstream_succ_nth_eq : IntFractPair.stream v (n + 1) = some ifp\ns_nth_eq : Stream'.Seq.get? (of v).s n = some { a := 1, b := ↑ifp.b }\n⊢ ∃ ifp_1, IntFractPair.stream v (n + 1) = some ifp_1 ∧ ↑ifp_1.b = { a := 1, b := ↑ifp.b }.b",
"tactic": "simp_all only [Option.some.injEq, exists_eq_left']"
},
{
"state_after": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq :\n Stream'.Seq.get?\n (match\n (IntFractPair.of v,\n Stream'.Seq.tail\n { val := IntFractPair.stream v, property := (_ : Stream'.IsSeq (IntFractPair.stream v)) }) with\n | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).s\n n =\n some gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ { a := 1, b := ↑ifp.b } = gp_n",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq : Stream'.Seq.get? (of v).s n = some gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ { a := 1, b := ↑ifp.b } = gp_n",
"tactic": "unfold of IntFractPair.seq1 at s_nth_eq"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\ngp_n : Pair K\ns_nth_eq :\n Stream'.Seq.get?\n (match\n (IntFractPair.of v,\n Stream'.Seq.tail\n { val := IntFractPair.stream v, property := (_ : Stream'.IsSeq (IntFractPair.stream v)) }) with\n | (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).s\n n =\n some gp_n\n⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ { a := 1, b := ↑ifp.b } = gp_n",
"tactic": "simpa [Stream'.Seq.get?_tail, Stream'.Seq.map_get?] using s_nth_eq"
}
] |
[
237,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Fin.succAbove_cycleRange
|
[
{
"state_after": "case zero\ni j : Fin Nat.zero\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)\n\ncase succ\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"state_before": "n : ℕ\ni j : Fin n\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "cases n"
},
{
"state_after": "case succ.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)\n\ncase succ.inr.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)\n\ncase succ.inr.inr\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"state_before": "case succ\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "rcases lt_trichotomy j i with (hlt | heq | hgt)"
},
{
"state_after": "no goals",
"state_before": "case zero\ni j : Fin Nat.zero\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "rcases j with ⟨_, ⟨⟩⟩"
},
{
"state_after": "case succ.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"state_before": "case succ.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "have : castSucc (j + 1) = j.succ := by\n ext\n rw [coe_castSucc, val_succ, Fin.val_add_one_of_lt (lt_of_lt_of_le hlt i.le_last)]"
},
{
"state_after": "case succ.inl.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ succ j ≠ 0\n\ncase succ.inl.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ succ j ≠ succ i\n\ncase succ.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑castSucc (j + 1) < succ i",
"state_before": "case succ.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "rw [Fin.cycleRange_of_lt hlt, Fin.succAbove_below, this, swap_apply_of_ne_of_ne]"
},
{
"state_after": "case h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\n⊢ ↑(↑castSucc (j + 1)) = ↑(succ j)",
"state_before": "n✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\n⊢ ↑castSucc (j + 1) = succ j",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\n⊢ ↑(↑castSucc (j + 1)) = ↑(succ j)",
"tactic": "rw [coe_castSucc, val_succ, Fin.val_add_one_of_lt (lt_of_lt_of_le hlt i.le_last)]"
},
{
"state_after": "no goals",
"state_before": "case succ.inl.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ succ j ≠ 0",
"tactic": "apply Fin.succ_ne_zero"
},
{
"state_after": "no goals",
"state_before": "case succ.inl.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ succ j ≠ succ i",
"tactic": "exact (Fin.succ_injective _).ne hlt.ne"
},
{
"state_after": "case succ.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑(↑castSucc (j + 1)) < ↑(succ i)",
"state_before": "case succ.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑castSucc (j + 1) < succ i",
"tactic": "rw [Fin.lt_iff_val_lt_val]"
},
{
"state_after": "no goals",
"state_before": "case succ.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhlt : j < i\nthis : ↑castSucc (j + 1) = succ j\n⊢ ↑(↑castSucc (j + 1)) < ↑(succ i)",
"tactic": "simpa [this] using hlt"
},
{
"state_after": "case succ.inr.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ ↑castSucc 0 < succ i",
"state_before": "case succ.inr.inl\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "rw [heq, Fin.cycleRange_self, Fin.succAbove_below, swap_apply_right, Fin.castSucc_zero]"
},
{
"state_after": "case succ.inr.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ 0 < succ i",
"state_before": "case succ.inr.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ ↑castSucc 0 < succ i",
"tactic": "rw [Fin.castSucc_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.inl.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nheq : j = i\n⊢ 0 < succ i",
"tactic": "apply Fin.succ_pos"
},
{
"state_after": "case succ.inr.inr.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ j ≠ 0\n\ncase succ.inr.inr.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ j ≠ succ i\n\ncase succ.inr.inr.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ i ≤ ↑castSucc j",
"state_before": "case succ.inr.inr\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ ↑(succAbove (succ i)) (↑(cycleRange i) j) = ↑(swap 0 (succ i)) (succ j)",
"tactic": "rw [Fin.cycleRange_of_gt hgt, Fin.succAbove_above, swap_apply_of_ne_of_ne]"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.inr.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ j ≠ 0",
"tactic": "apply Fin.succ_ne_zero"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.inr.a\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ j ≠ succ i",
"tactic": "apply (Fin.succ_injective _).ne hgt.ne.symm"
},
{
"state_after": "no goals",
"state_before": "case succ.inr.inr.h\nn✝ : ℕ\ni j : Fin (Nat.succ n✝)\nhgt : i < j\n⊢ succ i ≤ ↑castSucc j",
"tactic": "simpa [Fin.le_iff_val_le_val] using hgt"
}
] |
[
273,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivWithinAt.differentiableWithinAt
|
[] |
[
386,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Analysis/Convex/Slope.lean
|
ConcaveOn.slope_anti_adjacent
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) ≤ -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"tactic": "have := neg_le_neg (ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz)"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : -(((-f) z - (-f) y) / (z - y)) ≤ -(((-f) y - (-f) x) / (y - x))\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"tactic": "simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"tactic": "exact this"
}
] |
[
61,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.toFinsupp_apply
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\ni : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑{ toFinsupp := toFinsupp✝ }.toFinsupp i = coeff { toFinsupp := toFinsupp✝ } i",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q f : R[X]\ni : ℕ\n⊢ ↑f.toFinsupp i = coeff f i",
"tactic": "cases f"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\ni : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑{ toFinsupp := toFinsupp✝ }.toFinsupp i = coeff { toFinsupp := toFinsupp✝ } i",
"tactic": "rfl"
}
] |
[
668,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.map_le_restrict_range
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.120092\nR : Type ?u.120095\nR' : Type ?u.120098\nms : Set (OuterMeasure α)\nm : OuterMeasure α\nβ : Type u_1\nma : OuterMeasure α\nmb : OuterMeasure β\nf : α → β\nh : ↑(map f) ma ≤ mb\ns : Set β\n⊢ ↑(↑(map f) ma) s ≤ ↑(↑(restrict (range f)) mb) s",
"tactic": "simpa using h (s ∩ range f)"
}
] |
[
608,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
circle.injective_arg
|
[] |
[
30,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂_mul_monomial
|
[
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (↑C a * ↑(monomial s) a_1) = eval₂ f g (↑C a) * ↑f a_1 * Finsupp.prod s fun n e => g n ^ e\n\ncase h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (p q : MvPolynomial σ R),\n (∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n (∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g (q * ↑(monomial s) a) = eval₂ f g q * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g ((p + q) * ↑(monomial s) a) = eval₂ f g (p + q) * ↑f a * Finsupp.prod s fun n e => g n ^ e\n\ncase h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n (∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g (p * X n * ↑(monomial s) a) = eval₂ f g (p * X n) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "apply MvPolynomial.induction_on p"
},
{
"state_after": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a'✝ a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\na' : R\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (↑C a' * ↑(monomial s) a) = eval₂ f g (↑C a') * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"state_before": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (↑C a * ↑(monomial s) a_1) = eval₂ f g (↑C a) * ↑f a_1 * Finsupp.prod s fun n e => g n ^ e",
"tactic": "intro a' s a"
},
{
"state_after": "no goals",
"state_before": "case h_C\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a'✝ a₁ a₂ : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\na' : R\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (↑C a' * ↑(monomial s) a) = eval₂ f g (↑C a') * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "simp [C_mul_monomial, eval₂_monomial, f.map_mul]"
},
{
"state_after": "case h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np q : MvPolynomial σ R\nih_p : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\nih_q : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (q * ↑(monomial s) a) = eval₂ f g q * ↑f a * Finsupp.prod s fun n e => g n ^ e\n⊢ ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g ((p + q) * ↑(monomial s) a) = eval₂ f g (p + q) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"state_before": "case h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (p q : MvPolynomial σ R),\n (∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n (∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g (q * ↑(monomial s) a) = eval₂ f g q * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g ((p + q) * ↑(monomial s) a) = eval₂ f g (p + q) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "intro p q ih_p ih_q"
},
{
"state_after": "no goals",
"state_before": "case h_add\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q✝ : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np q : MvPolynomial σ R\nih_p : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\nih_q : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (q * ↑(monomial s) a) = eval₂ f g q * ↑f a * Finsupp.prod s fun n e => g n ^ e\n⊢ ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g ((p + q) * ↑(monomial s) a) = eval₂ f g (p + q) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "simp [add_mul, eval₂_add, ih_p, ih_q]"
},
{
"state_after": "case h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : σ\nih : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (p * X n * ↑(monomial s) a) = eval₂ f g (p * X n) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n (∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e) →\n ∀ {s : σ →₀ ℕ} {a : R},\n eval₂ f g (p * X n * ↑(monomial s) a) = eval₂ f g (p * X n) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "intro p n ih s a"
},
{
"state_after": "no goals",
"state_before": "case h_X\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : σ\nih : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (p * X n * ↑(monomial s) a) = eval₂ f g (p * X n) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "exact\n calc\n (p * X n * monomial s a).eval₂ f g = (p * monomial (Finsupp.single n 1 + s) a).eval₂ f g :=\n by rw [monomial_single_add, pow_one, mul_assoc]\n _ = (p * monomial (Finsupp.single n 1) 1).eval₂ f g * f a * s.prod fun n e => g n ^ e := by\n simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n f.map_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : σ\nih : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (p * X n * ↑(monomial s) a) = eval₂ f g (p * ↑(monomial (Finsupp.single n 1 + s)) a)",
"tactic": "rw [monomial_single_add, pow_one, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\nn✝ m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\np : MvPolynomial σ R\nn : σ\nih : ∀ {s : σ →₀ ℕ} {a : R}, eval₂ f g (p * ↑(monomial s) a) = eval₂ f g p * ↑f a * Finsupp.prod s fun n e => g n ^ e\ns : σ →₀ ℕ\na : R\n⊢ eval₂ f g (p * ↑(monomial (Finsupp.single n 1 + s)) a) =\n eval₂ f g (p * ↑(monomial (Finsupp.single n 1)) 1) * ↑f a * Finsupp.prod s fun n e => g n ^ e",
"tactic": "simp [ih, prod_single_index, prod_add_index, pow_one, pow_add, mul_assoc, mul_left_comm,\n f.map_one]"
}
] |
[
995,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.inv_iff
|
[
{
"state_after": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\n⊢ IsPrimitiveRoot ζ⁻¹ k → IsPrimitiveRoot ζ k",
"state_before": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\n⊢ IsPrimitiveRoot ζ⁻¹ k ↔ IsPrimitiveRoot ζ k",
"tactic": "refine' ⟨_, fun h => inv h⟩"
},
{
"state_after": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ⁻¹ k\n⊢ IsPrimitiveRoot ζ k",
"state_before": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\n⊢ IsPrimitiveRoot ζ⁻¹ k → IsPrimitiveRoot ζ k",
"tactic": "intro h"
},
{
"state_after": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ⁻¹ k\n⊢ IsPrimitiveRoot ζ⁻¹⁻¹ k",
"state_before": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ⁻¹ k\n⊢ IsPrimitiveRoot ζ k",
"tactic": "rw [← inv_inv ζ]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.2707095\nN : Type ?u.2707098\nG : Type u_1\nR : Type ?u.2707104\nS : Type ?u.2707107\nF : Type ?u.2707110\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ⁻¹ k\n⊢ IsPrimitiveRoot ζ⁻¹⁻¹ k",
"tactic": "exact inv h"
}
] |
[
581,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.exists_prod_set
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3341\nδ : Type ?u.3344\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\np : α × β → Prop\n⊢ (∃ x, x ∈ s ×ˢ t ∧ p x) ↔ ∃ x, x ∈ s ∧ ∃ y, y ∈ t ∧ p (x, y)",
"tactic": "simp [and_assoc]"
}
] |
[
107,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/CategoryTheory/Sites/Canonical.lean
|
CategoryTheory.Sheaf.le_finestTopology
|
[
{
"state_after": "case intro.mk.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX✝ Y : C\nS✝ : Sieve X✝\nR : Presieve X✝\nJ✝ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nJ : GrothendieckTopology C\nhJ : ∀ (P : Cᵒᵖ ⥤ Type v), P ∈ Ps → Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J X\nP : Cᵒᵖ ⥤ Type v\nhP : P ∈ Ps\n⊢ S ∈\n (fun f => ↑f X)\n { val := (finestTopologySingle P).sieves,\n property := (_ : ∃ a, a ∈ finestTopologySingle '' Ps ∧ a.sieves = (finestTopologySingle P).sieves) }",
"state_before": "C : Type u\ninst✝ : Category C\nP : Cᵒᵖ ⥤ Type v\nX Y : C\nS : Sieve X\nR : Presieve X\nJ✝ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nJ : GrothendieckTopology C\nhJ : ∀ (P : Cᵒᵖ ⥤ Type v), P ∈ Ps → Presieve.IsSheaf J P\n⊢ J ≤ finestTopology Ps",
"tactic": "rintro X S hS _ ⟨⟨_, _, ⟨P, hP, rfl⟩, rfl⟩, rfl⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX✝ Y✝ : C\nS✝ : Sieve X✝\nR : Presieve X✝\nJ✝ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nJ : GrothendieckTopology C\nhJ : ∀ (P : Cᵒᵖ ⥤ Type v), P ∈ Ps → Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J X\nP : Cᵒᵖ ⥤ Type v\nhP : P ∈ Ps\nY : C\nf : Y ⟶ X\n⊢ Presieve.IsSheafFor P (Sieve.pullback f S).arrows",
"state_before": "case intro.mk.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX✝ Y : C\nS✝ : Sieve X✝\nR : Presieve X✝\nJ✝ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nJ : GrothendieckTopology C\nhJ : ∀ (P : Cᵒᵖ ⥤ Type v), P ∈ Ps → Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J X\nP : Cᵒᵖ ⥤ Type v\nhP : P ∈ Ps\n⊢ S ∈\n (fun f => ↑f X)\n { val := (finestTopologySingle P).sieves,\n property := (_ : ∃ a, a ∈ finestTopologySingle '' Ps ∧ a.sieves = (finestTopologySingle P).sieves) }",
"tactic": "intro Y f"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nP✝ : Cᵒᵖ ⥤ Type v\nX✝ Y✝ : C\nS✝ : Sieve X✝\nR : Presieve X✝\nJ✝ J₂ : GrothendieckTopology C\nPs : Set (Cᵒᵖ ⥤ Type v)\nJ : GrothendieckTopology C\nhJ : ∀ (P : Cᵒᵖ ⥤ Type v), P ∈ Ps → Presieve.IsSheaf J P\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J X\nP : Cᵒᵖ ⥤ Type v\nhP : P ∈ Ps\nY : C\nf : Y ⟶ X\n⊢ Presieve.IsSheafFor P (Sieve.pullback f S).arrows",
"tactic": "exact hJ P hP (S.pullback f) (J.pullback_stable f hS)"
}
] |
[
208,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.monomial_zero_one
|
[] |
[
438,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.210172\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh1 : Antiperiodic f c₁\nh2 : Antiperiodic f c₂\n⊢ Periodic f (c₁ - c₂)",
"tactic": "simpa only [sub_eq_add_neg] using h1.add h2.neg"
}
] |
[
514,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mul_mk'_eq_mk'_of_mul
|
[] |
[
405,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Topology/VectorBundle/Constructions.lean
|
Trivialization.continuousLinearEquivAt_prod
|
[
{
"state_after": "case h.h\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv : E₁ x × E₂ x\n⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) v =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n v",
"state_before": "𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\n⊢ continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx =\n ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet))",
"tactic": "ext v : 2"
},
{
"state_after": "case h.h.mk\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) (v₁, v₂) =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n (v₁, v₂)",
"state_before": "case h.h\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv : E₁ x × E₂ x\n⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) v =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n v",
"tactic": "obtain ⟨v₁, v₂⟩ := v"
},
{
"state_after": "case h.h.mk\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (fun y =>\n (↑{\n toLocalHomeomorph :=\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) },\n baseSet := e₁.baseSet ∩ e₂.baseSet, open_baseSet := (_ : IsOpen (e₁.baseSet ∩ e₂.baseSet)),\n source_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source),\n target_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target),\n proj_toFun :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source →\n (↑{\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }\n x).fst =\n (↑{\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }\n x).fst) }\n (totalSpaceMk x y)).snd)\n (v₁, v₂) =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n (v₁, v₂)",
"state_before": "case h.h.mk\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ ↑(continuousLinearEquivAt 𝕜 (prod e₁ e₂) x hx) (v₁, v₂) =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n (v₁, v₂)",
"tactic": "rw [(e₁.prod e₂).continuousLinearEquivAt_apply 𝕜, Trivialization.prod]"
},
{
"state_after": "no goals",
"state_before": "case h.h.mk\n𝕜 : Type u_6\nB : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_2\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_3\ninst✝¹³ : TopologicalSpace (TotalSpace E₁)\nF₂ : Type u_4\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_5\ninst✝¹⁰ : TopologicalSpace (TotalSpace E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (prod e₁ e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (fun y =>\n (↑{\n toLocalHomeomorph :=\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧ (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun := (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) },\n baseSet := e₁.baseSet ∩ e₂.baseSet, open_baseSet := (_ : IsOpen (e₁.baseSet ∩ e₂.baseSet)),\n source_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source),\n target_eq :=\n (_ :\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target =\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ : ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.target),\n proj_toFun :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈\n {\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ :\n ContinuousOn (Prod.invFun' e₁ e₂)\n ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }.toLocalEquiv.source →\n (↑{\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }\n x).fst =\n (↑{\n toLocalEquiv :=\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ → x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) },\n open_source :=\n (_ :\n IsOpen\n { toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet),\n target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n map_source' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n (Prod.toFun' e₁ e₂ x).fst ∈ e₁.baseSet ∩ e₂.baseSet ∧\n (Prod.toFun' e₁ e₂ x).snd ∈ univ),\n map_target' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n x.fst ∈ e₁.baseSet ∩ e₂.baseSet),\n left_inv' :=\n (_ :\n ∀ (x : TotalSpace fun x => E₁ x × E₂ x),\n x ∈ TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet) →\n Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x),\n right_inv' :=\n (_ :\n ∀ (x : B × F₁ × F₂),\n x ∈ (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ →\n Prod.toFun' e₁ e₂ (Prod.invFun' e₁ e₂ x) = x) }.source),\n open_target := (_ : IsOpen ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)),\n continuous_toFun :=\n (_ :\n ContinuousOn (Prod.toFun' e₁ e₂) (TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet))),\n continuous_invFun :=\n (_ : ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ)) }\n x).fst) }\n (totalSpaceMk x y)).snd)\n (v₁, v₂) =\n ↑(ContinuousLinearEquiv.prod (continuousLinearEquivAt 𝕜 e₁ x (_ : x ∈ e₁.baseSet))\n (continuousLinearEquivAt 𝕜 e₂ x (_ : x ∈ e₂.baseSet)))\n (v₁, v₂)",
"tactic": "exact (congr_arg Prod.snd (prod_apply 𝕜 hx.1 hx.2 v₁ v₂) : _)"
}
] |
[
162,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/CategoryTheory/Preadditive/Mat.lean
|
CategoryTheory.Mat_.equivalenceSelfOfHasFiniteBiproducts_inverse
|
[] |
[
542,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.comp_addHom
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nf g✝ : α → β\nc c₁ c₂ x✝ : α\ninst✝¹ : Add α\ninst✝ : Add γ\nh : Periodic f c\ng : AddHom γ α\ng_inv : α → γ\nhg : RightInverse g_inv ↑g\nx : γ\n⊢ (f ∘ ↑g) (x + g_inv c) = (f ∘ ↑g) x",
"tactic": "simp only [hg c, h (g x), map_add, comp_apply]"
}
] |
[
66,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.le_graph_iff
|
[] |
[
908,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
907,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.le_of_not_lt
|
[] |
[
384,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
384,
11
] |
Mathlib/Data/Fin/Tuple/Monotone.lean
|
StrictMono.vecCons
|
[] |
[
73,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_sum
|
[] |
[
276,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Topology/Inseparable.lean
|
specializes_iff_pure
|
[] |
[
106,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.range_single_subset
|
[] |
[
374,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/CategoryTheory/Sites/Closed.lean
|
CategoryTheory.le_topology_of_closedSieves_isSheaf
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ S ∈ GrothendieckTopology.sieves J₂ X",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\n⊢ J₁ ≤ J₂",
"tactic": "intro X S hS"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ S ∈ GrothendieckTopology.sieves J₂ X",
"tactic": "rw [← J₂.close_eq_top_iff_mem]"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"tactic": "have : J₂.IsClosed (⊤ : Sieve X) := by\n intro Y f _\n trivial"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n { val := ⊤, property := this }",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"tactic": "suffices (⟨J₂.close S, J₂.close_isClosed S⟩ : Subtype _) = ⟨⊤, this⟩ by\n rw [Subtype.ext_iff] at this\n exact this"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n { val := ⊤, property := this }",
"tactic": "apply (h S hS).isSeparatedFor.ext"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nY : C\nf : Y ⟶ X\na✝ : GrothendieckTopology.Covers J₂ ⊤ f\n⊢ ⊤.arrows f",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\n⊢ GrothendieckTopology.IsClosed J₂ ⊤",
"tactic": "intro Y f _"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nY : C\nf : Y ⟶ X\na✝ : GrothendieckTopology.Covers J₂ ⊤ f\n⊢ ⊤.arrows f",
"tactic": "trivial"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis✝ : GrothendieckTopology.IsClosed J₂ ⊤\nthis :\n ↑{ val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n ↑{ val := ⊤, property := this✝ }\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis✝ : GrothendieckTopology.IsClosed J₂ ⊤\nthis :\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n { val := ⊤, property := this✝ }\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"tactic": "rw [Subtype.ext_iff] at this"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis✝ : GrothendieckTopology.IsClosed J₂ ⊤\nthis :\n ↑{ val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n ↑{ val := ⊤, property := this✝ }\n⊢ GrothendieckTopology.close J₂ S = ⊤",
"tactic": "exact this"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\n⊢ ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄,\n S.arrows f →\n (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"tactic": "intro Y f hf"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"tactic": "simp only [Functor.closedSieves_obj]"
},
{
"state_after": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑((Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) }) =\n ↑((Functor.closedSieves J₂).map f.op { val := ⊤, property := this })",
"state_before": "C : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ (Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) } =\n (Functor.closedSieves J₂).map f.op { val := ⊤, property := this }",
"tactic": "ext1"
},
{
"state_after": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f (GrothendieckTopology.close J₂ S) = Sieve.pullback f ⊤",
"state_before": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ↑((Functor.closedSieves J₂).map f.op\n { val := GrothendieckTopology.close J₂ S,\n property := (_ : GrothendieckTopology.IsClosed J₂ (GrothendieckTopology.close J₂ S)) }) =\n ↑((Functor.closedSieves J₂).map f.op { val := ⊤, property := this })",
"tactic": "dsimp"
},
{
"state_after": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ⊤ ∈ GrothendieckTopology.sieves J₂ Y",
"state_before": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ Sieve.pullback f (GrothendieckTopology.close J₂ S) = Sieve.pullback f ⊤",
"tactic": "rw [Sieve.pullback_top, ← J₂.pullback_close, S.pullback_eq_top_of_mem hf,\n J₂.close_eq_top_iff_mem]"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u\ninst✝ : Category C\nJ₁✝ J₂✝ J₁ J₂ : GrothendieckTopology C\nh : Presieve.IsSheaf J₁ (Functor.closedSieves J₂)\nX : C\nS : Sieve X\nhS : S ∈ GrothendieckTopology.sieves J₁ X\nthis : GrothendieckTopology.IsClosed J₂ ⊤\nY : C\nf : Y ⟶ X\nhf : S.arrows f\n⊢ ⊤ ∈ GrothendieckTopology.sieves J₂ Y",
"tactic": "apply J₂.top_mem"
}
] |
[
270,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_sub
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ exp (x - y) = exp x / exp y",
"tactic": "simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]"
}
] |
[
1173,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1172,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Prod.lean
|
QuadraticForm.anisotropic_of_pi
|
[
{
"state_after": "ι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\n⊢ ∀ (i : ι), Anisotropic (Q i)",
"state_before": "ι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : Anisotropic (pi Q)\n⊢ ∀ (i : ι), Anisotropic (Q i)",
"tactic": "simp_rw [Anisotropic, pi_apply, Function.funext_iff, Pi.zero_apply] at h"
},
{
"state_after": "ι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\ni : ι\nx : Mᵢ i\nhx : ↑(Q i) x = 0\n⊢ x = 0",
"state_before": "ι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\n⊢ ∀ (i : ι), Anisotropic (Q i)",
"tactic": "intro i x hx"
},
{
"state_after": "case refine_2\nι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\ni : ι\nx : Mᵢ i\nhx : ↑(Q i) x = 0\nthis : Pi.single i x i = 0\n⊢ x = 0\n\ncase refine_1\nι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\ni : ι\nx : Mᵢ i\nhx : ↑(Q i) x = 0\n⊢ ∑ i_1 : ι, ↑(Q i_1) (Pi.single i x i_1) = 0",
"state_before": "ι : Type u_1\nR✝ : Type ?u.66145\nM₁ : Type ?u.66148\nM₂ : Type ?u.66151\nN₁ : Type ?u.66154\nN₂ : Type ?u.66157\nMᵢ : ι → Type u_3\nNᵢ : ι → Type ?u.66167\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Module R✝ M₁\ninst✝⁹ : Module R✝ M₂\ninst✝⁸ : Module R✝ N₁\ninst✝⁷ : Module R✝ N₂\ninst✝⁶ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝⁵ : (i : ι) → AddCommMonoid (Nᵢ i)\ninst✝⁴ : (i : ι) → Module R✝ (Mᵢ i)\ninst✝³ : (i : ι) → Module R✝ (Nᵢ i)\ninst✝² : Fintype ι\nR : Type u_2\ninst✝¹ : OrderedRing R\ninst✝ : (i : ι) → Module R (Mᵢ i)\nQ : (i : ι) → QuadraticForm R (Mᵢ i)\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i : ι, ↑(Q i) (x i) = 0 → ∀ (a : ι), x a = 0\ni : ι\nx : Mᵢ i\nhx : ↑(Q i) x = 0\n⊢ x = 0",
"tactic": "have := h (Pi.single i x) ?_ i"
},
{
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{
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"tactic": "rw [Pi.single_eq_same, hx]"
},
{
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"tactic": "rw [Pi.single_eq_of_ne hji, map_zero]"
}
] |
[
165,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_le_iSup_of_subset
|
[] |
[
1442,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1441,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
Module.End.hasGeneralizedEigenvalue_of_hasEigenvalue
|
[
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : HasEigenvalue f μ\n⊢ HasGeneralizedEigenvalue f μ (Nat.succ 0)",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : HasEigenvalue f μ\n⊢ HasGeneralizedEigenvalue f μ k",
"tactic": "apply hasGeneralizedEigenvalue_of_hasGeneralizedEigenvalue_of_le hk"
},
{
"state_after": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : HasEigenvalue f μ\n⊢ LinearMap.ker (f - ↑(algebraMap R (End R M)) μ) ≠ ⊥",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : HasEigenvalue f μ\n⊢ HasGeneralizedEigenvalue f μ (Nat.succ 0)",
"tactic": "rw [HasGeneralizedEigenvalue, generalizedEigenspace, OrderHom.coe_fun_mk, pow_one]"
},
{
"state_after": "no goals",
"state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : HasEigenvalue f μ\n⊢ LinearMap.ker (f - ↑(algebraMap R (End R M)) μ) ≠ ⊥",
"tactic": "exact hμ"
}
] |
[
353,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
sdiff_unique
|
[
{
"state_after": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : x ⊓ y ⊔ z = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = ⊥\n⊢ x \\ y = z",
"state_before": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : x ⊓ y ⊔ z = x\ni : x ⊓ y ⊓ z = ⊥\n⊢ x \\ y = z",
"tactic": "conv_rhs at s => rw [← sup_inf_sdiff x y, sup_comm]"
},
{
"state_after": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = ⊥\n⊢ x \\ y = z",
"state_before": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : x ⊓ y ⊔ z = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = ⊥\n⊢ x \\ y = z",
"tactic": "rw [sup_comm] at s"
},
{
"state_after": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = x \\ y ⊓ (x ⊓ y)\n⊢ x \\ y = z",
"state_before": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = ⊥\n⊢ x \\ y = z",
"tactic": "conv_rhs at i => rw [← inf_inf_sdiff x y, inf_comm]"
},
{
"state_after": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : z ⊓ (x ⊓ y) = x \\ y ⊓ (x ⊓ y)\n⊢ x \\ y = z",
"state_before": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : x ⊓ y ⊓ z = x \\ y ⊓ (x ⊓ y)\n⊢ x \\ y = z",
"tactic": "rw [inf_comm] at i"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.2138\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\ns : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y\ni : z ⊓ (x ⊓ y) = x \\ y ⊓ (x ⊓ y)\n⊢ x \\ y = z",
"tactic": "exact (eq_of_inf_eq_sup_eq i s).symm"
}
] |
[
135,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.comap_id'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU : Opens ↑(PrimeSpectrum.Top R)\np : ↑(PrimeSpectrum.Top R)\nhpU : p ∈ U.carrier\n⊢ p ∈ ↑(PrimeSpectrum.comap (RingHom.id R)) ⁻¹' U.carrier",
"tactic": "rwa [PrimeSpectrum.comap_id]"
},
{
"state_after": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ eqToHom (_ : (structureSheaf R).val.obj U.op = (structureSheaf R).val.obj U.op) =\n RingHom.id ↑((structureSheaf R).val.obj U.op)",
"state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ comap (RingHom.id R) U U\n (_ : ∀ (p : ↑(PrimeSpectrum.Top R)), p ∈ U.carrier → p ∈ ↑(PrimeSpectrum.comap (RingHom.id R)) ⁻¹' U.carrier) =\n RingHom.id ↑((structureSheaf R).val.obj U.op)",
"tactic": "rw [comap_id U U rfl]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝² : CommRing R\nS : Type u\ninst✝¹ : CommRing S\nP : Type u\ninst✝ : CommRing P\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ eqToHom (_ : (structureSheaf R).val.obj U.op = (structureSheaf R).val.obj U.op) =\n RingHom.id ↑((structureSheaf R).val.obj U.op)",
"tactic": "rfl"
}
] |
[
1196,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1194,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.mem_replicate
|
[] |
[
918,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
917,
1
] |
Mathlib/Order/Hom/Set.lean
|
OrderIso.symm_preimage_preimage
|
[] |
[
55,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.tr_eval
|
[
{
"state_after": "case intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝ : b₁ ∈ eval f₁ a₁\nab : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\n⊢ ∃ b₂, tr b₁ b₂ ∧ b₂ ∈ eval f₂ a₂",
"state_before": "σ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab : b₁ ∈ eval f₁ a₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ b₂ ∈ eval f₂ a₂",
"tactic": "cases' mem_eval.1 ab with ab b0"
},
{
"state_after": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\n⊢ ∃ b₂, tr b₁ b₂ ∧ b₂ ∈ eval f₂ a₂",
"state_before": "case intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝ : b₁ ∈ eval f₁ a₁\nab : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\n⊢ ∃ b₂, tr b₁ b₂ ∧ b₂ ∈ eval f₂ a₂",
"tactic": "rcases tr_reaches H aa ab with ⟨b₂, bb, ab⟩"
},
{
"state_after": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\n⊢ f₂ b₂ = none",
"state_before": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\n⊢ ∃ b₂, tr b₁ b₂ ∧ b₂ ∈ eval f₂ a₂",
"tactic": "refine' ⟨_, bb, mem_eval.2 ⟨ab, _⟩⟩"
},
{
"state_after": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\nthis :\n match f₁ b₁ with\n | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1\n | none => f₂ b₂ = none\n⊢ f₂ b₂ = none",
"state_before": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\n⊢ f₂ b₂ = none",
"tactic": "have := H bb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ b₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nab✝¹ : b₁ ∈ eval f₁ a₁\nab✝ : Reaches f₁ a₁ b₁\nb0 : f₁ b₁ = none\nb₂ : σ₂\nbb : tr b₁ b₂\nab : Reaches f₂ a₂ b₂\nthis :\n match f₁ b₁ with\n | some b₁ => ∃ b₂_1, tr b₁ b₂_1 ∧ Reaches₁ f₂ b₂ b₂_1\n | none => f₂ b₂ = none\n⊢ f₂ b₂ = none",
"tactic": "rwa [b0] at this"
}
] |
[
931,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
926,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.Ico_filter_le_left
|
[
{
"state_after": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ a b : α\ninst✝ : DecidablePred fun x => x ≤ a\nhab : a < b\n⊢ {a}.val = {a}",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ a b : α\ninst✝ : DecidablePred fun x => x ≤ a\nhab : a < b\n⊢ filter (fun x => x ≤ a) (Ico a b) = {a}",
"tactic": "rw [Ico, ← Finset.filter_val, Finset.Ico_filter_le_left hab]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ a b : α\ninst✝ : DecidablePred fun x => x ≤ a\nhab : a < b\n⊢ {a}.val = {a}",
"tactic": "rfl"
}
] |
[
223,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_pow'
|
[] |
[
179,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.equiv_symm
|
[] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.leftMoves_add
|
[] |
[
1476,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1475,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.Normal.mem_comm
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.339502\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.339511\ninst✝ : AddGroup A\nH K : Subgroup G\nnH : Normal H\na b : G\nh : a * b ∈ H\nthis : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ H\n⊢ b * a ∈ H",
"state_before": "G : Type u_1\nG' : Type ?u.339502\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.339511\ninst✝ : AddGroup A\nH K : Subgroup G\nnH : Normal H\na b : G\nh : a * b ∈ H\n⊢ b * a ∈ H",
"tactic": "have : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ H := nH.conj_mem (a * b) h a⁻¹"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.339502\ninst✝² : Group G\ninst✝¹ : Group G'\nA : Type ?u.339511\ninst✝ : AddGroup A\nH K : Subgroup G\nnH : Normal H\na b : G\nh : a * b ∈ H\nthis : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ H\n⊢ b * a ∈ H",
"tactic": "simp_all only [inv_mul_cancel_left, inv_inv]"
}
] |
[
1945,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1941,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cone.ofFork_π
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\nF : WalkingParallelPair ⥤ C\nt : Fork (F.map left) (F.map right)\nj : WalkingParallelPair\n⊢ (parallelPair (F.map left) (F.map right)).obj j = F.obj j",
"tactic": "aesop"
}
] |
[
624,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/RingTheory/Adjoin/PowerBasis.lean
|
PowerBasis.repr_mul_isIntegral
|
[
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n⊢ IsIntegral R (↑(↑B.basis.repr (x * y)) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\n⊢ ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr (x * y)) i)",
"tactic": "intro i"
},
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n⊢ IsIntegral R\n (Finset.sum (Finset.univ ×ˢ Finset.univ) fun k =>\n ↑(↑B.basis.repr (↑(↑B.basis.repr x) k.fst • ↑B.basis k.fst * ↑(↑B.basis.repr y) k.snd • ↑B.basis k.snd)) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n⊢ IsIntegral R (↑(↑B.basis.repr (x * y)) i)",
"tactic": "rw [← B.basis.sum_repr x, ← B.basis.sum_repr y, Finset.sum_mul_sum, LinearEquiv.map_sum,\n Finset.sum_apply']"
},
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R\n (↑(↑B.basis.repr (↑(↑B.basis.repr x) I.fst • ↑B.basis I.fst * ↑(↑B.basis.repr y) I.snd • ↑B.basis I.snd)) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n⊢ IsIntegral R\n (Finset.sum (Finset.univ ×ˢ Finset.univ) fun k =>\n ↑(↑B.basis.repr (↑(↑B.basis.repr x) k.fst • ↑B.basis k.fst * ↑(↑B.basis.repr y) k.snd • ↑B.basis k.snd)) i)",
"tactic": "refine' IsIntegral.sum _ fun I _ => _"
},
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R\n ((↑(↑B.basis.repr y) I.snd • ↑(↑B.basis.repr x) I.fst • ↑(↑B.basis.repr (↑B.basis I.fst * ↑B.basis I.snd))) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R\n (↑(↑B.basis.repr (↑(↑B.basis.repr x) I.fst • ↑B.basis I.fst * ↑(↑B.basis.repr y) I.snd • ↑B.basis I.snd)) i)",
"tactic": "simp only [Algebra.smul_mul_assoc, Algebra.mul_smul_comm, LinearEquiv.map_smulₛₗ,\n RingHom.id_apply, Finsupp.coe_smul, Pi.smul_apply, id.smul_eq_mul]"
},
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R (↑(↑B.basis.repr (↑B.basis I.fst * ↑B.basis I.snd)) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R\n ((↑(↑B.basis.repr y) I.snd • ↑(↑B.basis.repr x) I.fst • ↑(↑B.basis.repr (↑B.basis I.fst * ↑B.basis I.snd))) i)",
"tactic": "refine' isIntegral_mul (hy _) (isIntegral_mul (hx _) _)"
},
{
"state_after": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R (↑(↑B.basis.repr (B.gen ^ (↑I.fst + ↑I.snd))) i)",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R (↑(↑B.basis.repr (↑B.basis I.fst * ↑B.basis I.snd)) i)",
"tactic": "simp only [coe_basis, ← pow_add]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.143505\nS : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra K S\nR : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsScalarTower R K S\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst✝ : IsDomain S\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr x) i)\nhy : ∀ (i : Fin B.dim), IsIntegral R (↑(↑B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim × Fin B.dim\nx✝ : I ∈ Finset.univ ×ˢ Finset.univ\n⊢ IsIntegral R (↑(↑B.basis.repr (B.gen ^ (↑I.fst + ↑I.snd))) i)",
"tactic": "refine' repr_gen_pow_isIntegral hB hmin _ _"
}
] |
[
150,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
Inducing.continuousInv
|
[] |
[
413,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Logic/Equiv/Option.lean
|
Equiv.optionSubtype_symm_apply_apply_coe
|
[] |
[
243,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Data/List/Sublists.lean
|
List.revzip_sublists'
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ revzip (sublists' l) → l₁ ++ l₂ ~ l",
"tactic": "rw [revzip]"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nh : (l₁, l₂) ∈ zip (sublists' []) (reverse (sublists' []))\n⊢ l₁ ++ l₂ ~ []\n\ncase cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂ : List α\nh : (l₁, l₂) ∈ zip (sublists' (a :: l)) (reverse (sublists' (a :: l)))\n⊢ l₁ ++ l₂ ~ a :: l",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl : List α\n⊢ ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l",
"tactic": "induction' l with a l IH <;> intro l₁ l₂ h"
},
{
"state_after": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nh : l₁ = [] ∧ l₂ = []\n⊢ l₁ ++ l₂ ~ []",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nh : (l₁, l₂) ∈ zip (sublists' []) (reverse (sublists' []))\n⊢ l₁ ++ l₂ ~ []",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nh : l₁ = [] ∧ l₂ = []\n⊢ l₁ ++ l₂ ~ []",
"tactic": "simp [h]"
},
{
"state_after": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂ : List α\nh :\n (∃ a_1 b, (a_1, b) ∈ zip (sublists' l) (reverse (sublists' l)) ∧ a_1 = l₁ ∧ a :: b = l₂) ∨\n ∃ a_1, (a_1, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) ∧ a :: a_1 = l₁\n⊢ l₁ ++ l₂ ~ a :: l",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂ : List α\nh : (l₁, l₂) ∈ zip (sublists' (a :: l)) (reverse (sublists' (a :: l)))\n⊢ l₁ ++ l₂ ~ a :: l",
"tactic": "rw [sublists'_cons, reverse_append, zip_append, ← map_reverse, zip_map_right, zip_map_left] at *\n <;> [simp at h; simp]"
},
{
"state_after": "case cons.inl.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂' : List α\nh : (l₁, l₂') ∈ zip (sublists' l) (reverse (sublists' l))\n⊢ l₁ ++ a :: l₂' ~ a :: l\n\ncase cons.inr.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₂ l₁' : List α\nh : (l₁', l₂) ∈ zip (sublists' l) (reverse (sublists' l))\n⊢ a :: l₁' ++ l₂ ~ a :: l",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂ : List α\nh :\n (∃ a_1 b, (a_1, b) ∈ zip (sublists' l) (reverse (sublists' l)) ∧ a_1 = l₁ ∧ a :: b = l₂) ∨\n ∃ a_1, (a_1, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) ∧ a :: a_1 = l₁\n⊢ l₁ ++ l₂ ~ a :: l",
"tactic": "rcases h with (⟨l₁, l₂', h, rfl, rfl⟩ | ⟨l₁', h, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case cons.inl.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₁ l₂' : List α\nh : (l₁, l₂') ∈ zip (sublists' l) (reverse (sublists' l))\n⊢ l₁ ++ a :: l₂' ~ a :: l",
"tactic": "exact perm_middle.trans ((IH _ _ h).cons _)"
},
{
"state_after": "no goals",
"state_before": "case cons.inr.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\na : α\nl : List α\nIH : ∀ (l₁ l₂ : List α), (l₁, l₂) ∈ zip (sublists' l) (reverse (sublists' l)) → l₁ ++ l₂ ~ l\nl₂ l₁' : List α\nh : (l₁', l₂) ∈ zip (sublists' l) (reverse (sublists' l))\n⊢ a :: l₁' ++ l₂ ~ a :: l",
"tactic": "exact (IH _ _ h).cons _"
}
] |
[
458,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.inl_snd
|
[] |
[
591,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
589,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.map_iff_of_equiv
|
[] |
[
283,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
11
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.integralSum_neg
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI J : Box ι\nπ✝ : TaggedPrepartition I\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\n⊢ integralSum (-f) vol π = -integralSum f vol π",
"tactic": "simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]"
}
] |
[
147,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.Nonempty.of_div_left
|
[] |
[
607,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
606,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.ofDual_lt_iff
|
[] |
[
1029,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1027,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_eq_top_iff_finite
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ 1 ∈ span s ↔ ∃ s', ↑s' ⊆ s ∧ 1 ∈ span ↑s'",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ span s = ⊤ ↔ ∃ s', ↑s' ⊆ s ∧ span ↑s' = ⊤",
"tactic": "simp_rw [eq_top_iff_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Semiring α\nI : Ideal α\na b : α\ns : Set α\n⊢ 1 ∈ span s ↔ ∃ s', ↑s' ⊆ s ∧ 1 ∈ span ↑s'",
"tactic": "exact ⟨Submodule.mem_span_finite_of_mem_span, fun ⟨s', h₁, h₂⟩ => span_mono h₁ h₂⟩"
}
] |
[
216,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.leadingCoeff_mul_X_pow
|
[] |
[
1021,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.exists_ultrafilter_iff
|
[] |
[
458,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Order/Lattice.lean
|
inf_assoc
|
[] |
[
503,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.isOpen_basicOpen
|
[] |
[
403,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Topology/Homotopy/Contractible.lean
|
id_nullhomotopic
|
[
{
"state_after": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : ContractibleSpace X\nhv : X ≃ₕ Unit\n⊢ Nullhomotopic (ContinuousMap.id X)",
"state_before": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : ContractibleSpace X\n⊢ Nullhomotopic (ContinuousMap.id X)",
"tactic": "obtain ⟨hv⟩ := ContractibleSpace.hequiv_unit X"
},
{
"state_after": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : ContractibleSpace X\nhv : X ≃ₕ Unit\n⊢ Homotopic (ContinuousMap.id X) (const X (↑hv.invFun ()))",
"state_before": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : ContractibleSpace X\nhv : X ≃ₕ Unit\n⊢ Nullhomotopic (ContinuousMap.id X)",
"tactic": "use hv.invFun ()"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : ContractibleSpace X\nhv : X ≃ₕ Unit\n⊢ Homotopic (ContinuousMap.id X) (const X (↑hv.invFun ()))",
"tactic": "convert hv.left_inv.symm"
}
] |
[
69,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.isAdjointPair_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝²⁰ : Semiring R\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : Module R M\nR₁ : Type ?u.1026415\nM₁ : Type ?u.1026418\ninst✝¹⁷ : Ring R₁\ninst✝¹⁶ : AddCommGroup M₁\ninst✝¹⁵ : Module R₁ M₁\nR₂ : Type ?u.1027027\nM₂ : Type ?u.1027030\ninst✝¹⁴ : CommSemiring R₂\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R₂ M₂\nR₃ : Type ?u.1027217\nM₃ : Type ?u.1027220\ninst✝¹¹ : CommRing R₃\ninst✝¹⁰ : AddCommGroup M₃\ninst✝⁹ : Module R₃ M₃\nV : Type ?u.1027808\nK : Type ?u.1027811\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1029025\nM₂'' : Type ?u.1029028\ninst✝⁵ : AddCommMonoid M₂'\ninst✝⁴ : AddCommMonoid M₂''\ninst✝³ : Module R₂ M₂'\ninst✝² : Module R₂ M₂''\nF : BilinForm R M\nM' : Type u_3\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB' : BilinForm R M'\nf f' : M →ₗ[R] M'\ng g' : M' →ₗ[R] M\nx : M\ny : M'\n⊢ bilin B' (↑0 x) y = bilin B x (↑0 y)",
"tactic": "simp only [BilinForm.zero_left, BilinForm.zero_right, LinearMap.zero_apply]"
}
] |
[
1043,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1042,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.coe_mul
|
[] |
[
1726,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1725,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_bot
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.56692\nα : Type u_1\nβ : Type ?u.56698\nπ : ι → Type ?u.56703\ninst✝ : HeytingAlgebra α\na : α\n⊢ a ⇔ ⊥ = aᶜ",
"tactic": "simp [bihimp]"
}
] |
[
369,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
Submodule.triorthogonal_eq_orthogonal
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.883149\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ topologicalClosure Kᗮ = Kᗮ",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.883149\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ Kᗮᗮᗮ = Kᗮ",
"tactic": "rw [Kᗮ.orthogonal_orthogonal_eq_closure]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.883149\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ topologicalClosure Kᗮ = Kᗮ",
"tactic": "exact K.isClosed_orthogonal.submodule_topologicalClosure_eq"
}
] |
[
927,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
925,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image_image₂_right_anticomm
|
[] |
[
505,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Analysis/Convex/Independent.lean
|
ConvexIndependent.subtype
|
[] |
[
95,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
11
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
MeasureTheory.MeasurablySeparable.iUnion
|
[
{
"state_after": "α✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurablySeparable (⋃ (n : ι), s n) (⋃ (m : ι), t m)",
"state_before": "α✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nh : ∀ (m n : ι), MeasurablySeparable (s m) (t n)\n⊢ MeasurablySeparable (⋃ (n : ι), s n) (⋃ (m : ι), t m)",
"tactic": "choose u hsu htu hu using h"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ (⋃ (n : ι), s n) ⊆ ⋃ (m : ι), ⋂ (n : ι), u m n\n\ncase refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ Disjoint (⋃ (m : ι), t m) (⋃ (m : ι), ⋂ (n : ι), u m n)\n\ncase refine'_3\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurableSet (⋃ (m : ι), ⋂ (n : ι), u m n)",
"state_before": "α✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurablySeparable (⋃ (n : ι), s n) (⋃ (m : ι), t m)",
"tactic": "refine' ⟨⋃ m, ⋂ n, u m n, _, _, _⟩"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nm : ι\n⊢ s m ⊆ ⋂ (n : ι), u m n",
"state_before": "case refine'_1\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ (⋃ (n : ι), s n) ⊆ ⋃ (m : ι), ⋂ (n : ι), u m n",
"tactic": "refine' iUnion_subset fun m => subset_iUnion_of_subset m _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nm : ι\n⊢ s m ⊆ ⋂ (n : ι), u m n",
"tactic": "exact subset_iInter fun n => hsu m n"
},
{
"state_after": "case refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ ∀ (i i_1 : ι), Disjoint (t i) (⋂ (n : ι), u i_1 n)",
"state_before": "case refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ Disjoint (⋃ (m : ι), t m) (⋃ (m : ι), ⋂ (n : ι), u m n)",
"tactic": "simp_rw [disjoint_iUnion_left, disjoint_iUnion_right]"
},
{
"state_after": "case refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nn m : ι\n⊢ Disjoint (t n) (⋂ (n : ι), u m n)",
"state_before": "case refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ ∀ (i i_1 : ι), Disjoint (t i) (⋂ (n : ι), u i_1 n)",
"tactic": "intro n m"
},
{
"state_after": "α✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nn m : ι\n⊢ (⋂ (n : ι), u m n) ≤ u m n",
"state_before": "case refine'_2\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nn m : ι\n⊢ Disjoint (t n) (⋂ (n : ι), u m n)",
"tactic": "apply Disjoint.mono_right _ (htu m n)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nn m : ι\n⊢ (⋂ (n : ι), u m n) ≤ u m n",
"tactic": "apply iInter_subset"
},
{
"state_after": "case refine'_3\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nm : ι\n⊢ MeasurableSet (⋂ (n : ι), u m n)",
"state_before": "case refine'_3\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\n⊢ MeasurableSet (⋃ (m : ι), ⋂ (n : ι), u m n)",
"tactic": "refine' MeasurableSet.iUnion fun m => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα✝ : Type ?u.67309\ninst✝² : TopologicalSpace α✝\nι : Type u_1\ninst✝¹ : Countable ι\nα : Type u_2\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nm : ι\n⊢ MeasurableSet (⋂ (n : ι), u m n)",
"tactic": "exact MeasurableSet.iInter fun n => hu m n"
}
] |
[
286,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
275,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformContinuous_toMul
|
[] |
[
1452,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1451,
1
] |
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