url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | rewrite [rep_common_image_step] at h6 | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | obtain (a : U) (h9 : a ∈ rep_common_image R S X0 m ∧
∃ (y : V), R a y ∧ S c y) from h6 | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | apply Exists.intro a | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | rewrite [invRel_def, Tdef] | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | apply Or.inl | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | obtain (y : V) (h10 : R a y ∧ S c y) from h9.right | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have h11 : y = b :=
fcnl_unique S_match_CB.right.left cC h10.right Scb | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | rewrite [h11] at h10 | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | apply And.intro _ h10.left | case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.h
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | define | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show ∃ (n : Nat), a ∈ rep_common_image R S X0 n from
Exists.intro m h9.left | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | by_contra h7 | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | rewrite [h7] at h6 | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | define at h6 | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show False from h6.right cC | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | apply Exists.intro c | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | rewrite [invRel_def, Tdef] | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show c ∈ X ∧ R c b ∨ c ∉ X ∧ S c b from
Or.inr (And.intro h5 Scb) | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | fix a1 : U | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | fix a2 : U | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | assume Ta1b : T a1 b | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | assume Ta2b : T a2 b | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | by_cases h5 : a1 ∈ X | case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case right.right.Uniqueness
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have h6 : a2 ∈ X := csb_cri_of_cri Ta1b Ta2b h5 | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have Ra1b : R a1 b := csb_match_cri Ta1b h5 | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have Ra2b : R a2 b := csb_match_cri Ta2b h6 | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have h7 : b ∈ D := (R_match_AD.left Ra1b).right | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show a1 = a2 from
fcnl_unique R_match_AD.right.right h7 Ra1b Ra2b | case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have h6 : a2 ∉ X := by
by_contra h6
show False from h5 (csb_cri_of_cri Ta2b Ta1b h6)
done | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have Sa1b : S a1 b := csb_match_not_cri Ta1b h5 | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | have Sa2b : S a2 b := csb_match_not_cri Ta2b h6 | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show a1 = a2 from
fcnl_unique S_match_CB.right.right bB Sa1b Sa2b | case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_m... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U :=... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | by_contra h6 | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap8Part2.lean | HTPI.Cantor_Schroeder_Bernstein_theorem | [1559, 1] | [1718, 7] | show False from h5 (csb_cri_of_cri Ta2b Ta1b h6) | U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X : Set U := cum_rep_image R S X0
T : Rel U V := csb_match R S ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
U V : Type
A C : Set U
B D : Set V
h1 : C ⊆ A
h2 : D ⊆ B
h3 : A ∼ D
h4 : C ∼ B
R : Rel U V
R_match_AD : rel_within R A D ∧ fcnl_on R A ∧ fcnl_on (invRel R) D
S : Rel U V
S_match_CB : rel_within S C B ∧ fcnl_on S C ∧ fcnl_on (invRel S) B
X0 : Set U := A \ C
X ... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.graph_def | [43, 1] | [44, 41] | rfl | A B : Type
f : A → B
a : A
b : B
⊢ (a, b) ∈ graph f ↔ f a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
f : A → B
a : A
b : B
⊢ (a, b) ∈ graph f ↔ f a = b
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | assume h1 : is_func_graph F | A B : Type
F : Set (A × B)
⊢ is_func_graph F → ∃ f, graph f = F | A B : Type
F : Set (A × B)
h1 : is_func_graph F
⊢ ∃ f, graph f = F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
⊢ is_func_graph F → ∃ f, graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | define at h1 | A B : Type
F : Set (A × B)
h1 : is_func_graph F
⊢ ∃ f, graph f = F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
⊢ ∃ f, graph f = F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : is_func_graph F
⊢ ∃ f, graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | have h2 : ∀ (x : A), ∃ (y : B), (x, y) ∈ F := by
fix x : A
obtain (y : B) (h3 : (x, y) ∈ F)
(h4 : ∀ (y1 y2 : B), (x, y1) ∈ F → (x, y2) ∈ F → y1 = y2) from h1 x
show ∃ (y : B), (x, y) ∈ F from Exists.intro y h3
done | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
⊢ ∃ f, graph f = F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
⊢ ∃ f, graph f = F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
⊢ ∃ f, graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | set f : A → B := fun (x : A) => Classical.choose (h2 x) | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
⊢ ∃ f, graph f = F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∃ f, graph f = F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
⊢ ∃ f, graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | apply Exists.intro f | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∃ f, graph f = F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ graph f = F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∃ f, graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | apply Set.ext | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ graph f = F | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∀ (x : A × B), x ∈ graph f ↔ x ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ graph f = F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | fix (x, y) : A × B | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∀ (x : A × B), x ∈ graph f ↔ x ∈ F | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
⊢ ∀ (x : A × B), x ∈ graph f ↔ x ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | have h3 : (x, f x) ∈ F := Classical.choose_spec (h2 x) | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | apply Iff.intro | case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f → (x, y) ∈ F
case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f ↔ (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | fix x : A | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
⊢ ∀ (x : A), ∃ y, (x, y) ∈ F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
⊢ ∃ y, (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
⊢ ∀ (x : A), ∃ y, (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | obtain (y : B) (h3 : (x, y) ∈ F)
(h4 : ∀ (y1 y2 : B), (x, y1) ∈ F → (x, y2) ∈ F → y1 = y2) from h1 x | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
⊢ ∃ y, (x, y) ∈ F | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
y : B
h3 : (x, y) ∈ F
h4 : ∀ (y_1 y_2 : B), (x, y_1) ∈ F → (x, y_2) ∈ F → y_1 = y_2
⊢ ∃ y, (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
⊢ ∃ y, (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | show ∃ (y : B), (x, y) ∈ F from Exists.intro y h3 | A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
y : B
h3 : (x, y) ∈ F
h4 : ∀ (y_1 y_2 : B), (x, y_1) ∈ F → (x, y_2) ∈ F → y_1 = y_2
⊢ ∃ y, (x, y) ∈ F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
x : A
y : B
h3 : (x, y) ∈ F
h4 : ∀ (y_1 y_2 : B), (x, y_1) ∈ F → (x, y_2) ∈ F → y_1 = y_2
⊢ ∃ y, (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | assume h4 : (x, y) ∈ graph f | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f → (x, y) ∈ F | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ graph f
⊢ (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ graph f → (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | define at h4 | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ graph f
⊢ (x, y) ∈ F | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ graph f
⊢ (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | rewrite [h4] at h3 | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, y) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | show (x, y) ∈ F from h3 | case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, y) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, y) ∈ F
h4 : f x = y
⊢ (x, y) ∈ F
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | assume h4 : (x, y) ∈ F | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ F → (x, y) ∈ graph f | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ (x, y) ∈ graph f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
⊢ (x, y) ∈ F → (x, y) ∈ graph f
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | define | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ (x, y) ∈ graph f | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ f x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ (x, y) ∈ graph f
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | obtain (z : B) (h5 : (x, z) ∈ F)
(h6 : ∀ (y1 y2 : B), (x, y1) ∈ F → (x, y2) ∈ F → y1 = y2) from h1 x | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ f x = y | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
z : B
h5 : (x, z) ∈ F
h6 : ∀ (y_1 y_2 : B), (x, y_1) ∈ F → (x, y_2) ∈ F → y_1 = y_2
⊢ f x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
⊢ f x = y
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.func_from_graph_rtl | [50, 1] | [79, 7] | show f x = y from h6 (f x) y h3 h4 | case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
z : B
h5 : (x, z) ∈ F
h6 : ∀ (y_1 y_2 : B), (x, y_1) ∈ F → (x, y_2) ∈ F → y_1 = y_2
⊢ f x = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B : Type
F : Set (A × B)
h1 : ∀ (x : A), ∃! y, (x, y) ∈ F
h2 : ∀ (x : A), ∃ y, (x, y) ∈ F
f : A → B := fun x => Classical.choose (_ : ∃ y, (x, y) ∈ F)
x : A
y : B
h3 : (x, f x) ∈ F
h4 : (x, y) ∈ F
z : B
h5 : (x, z) ∈ F
h6 : ∀ (y_1 y_2 : B), (x, y... |
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | set h : A → C := fun (x : A) => g (f x) | A B C : Type
f : A → B
g : B → C
⊢ ∃ h, graph h = comp (graph g) (graph f) | A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∃ h, graph h = comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
⊢ ∃ h, graph h = comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | apply Exists.intro h | A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∃ h, graph h = comp (graph g) (graph f) | A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ graph h = comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∃ h, graph h = comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | apply Set.ext | A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ graph h = comp (graph g) (graph f) | case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∀ (x : A × C), x ∈ graph h ↔ x ∈ comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ graph h = comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | fix (a, c) : A × C | case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∀ (x : A × C), x ∈ graph h ↔ x ∈ comp (graph g) (graph f) | case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h ↔ (a, c) ∈ comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
⊢ ∀ (x : A × C), x ∈ graph h ↔ x ∈ comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | apply Iff.intro | case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h ↔ (a, c) ∈ comp (graph g) (graph f) | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h → (a, c) ∈ comp (graph g) (graph f)
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ comp (graph g) (graph f) → (a, c) ∈ graph h | Please generate a tactic in lean4 to solve the state.
STATE:
case h
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h ↔ (a, c) ∈ comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | assume h1 : (a, c) ∈ graph h | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h → (a, c) ∈ comp (graph g) (graph f) | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ graph h
⊢ (a, c) ∈ comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ graph h → (a, c) ∈ comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define at h1 | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ graph h
⊢ (a, c) ∈ comp (graph g) (graph f) | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, c) ∈ comp (graph g) (graph f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ graph h
⊢ (a, c) ∈ comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, c) ∈ comp (graph g) (graph f) | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, c) ∈ comp (graph g) (graph f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | apply Exists.intro (f a) | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f ∧ (f a, c) ∈ graph g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | apply And.intro | case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f ∧ (f a, c) ∈ graph g | case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f
case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (f a, c) ∈ graph g | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f ∧ (f a, c) ∈ graph g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define | case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f | case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ f a = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (a, f a) ∈ graph f
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | rfl | case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ f a = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.left
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ f a = f a
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define | case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (f a, c) ∈ graph g | case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ g (f a) = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ (f a, c) ∈ graph g
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | show g (f a) = c from h1 | case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ g (f a) = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.right
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : h a = c
⊢ g (f a) = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | assume h1 : (a, c) ∈ comp (graph g) (graph f) | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ comp (graph g) (graph f) → (a, c) ∈ graph h | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ (a, c) ∈ graph h | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
⊢ (a, c) ∈ comp (graph g) (graph f) → (a, c) ∈ graph h
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ (a, c) ∈ graph h | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ (a, c) ∈ graph h
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define at h1 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : (a, c) ∈ comp (graph g) (graph f)
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | obtain (b : B) (h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g) from h1 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | have h3 : (a, b) ∈ graph f := h2.left | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | have h4 : (b, c) ∈ graph g := h2.right | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
h4 : (b, c) ∈ graph g
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define at h3 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
h4 : (b, c) ∈ graph g
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : (b, c) ∈ graph g
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : (a, b) ∈ graph f
h4 : (b, c) ∈ graph g
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | define at h4 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : (b, c) ∈ graph g
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g b = c
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : (b, c) ∈ graph g
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | rewrite [←h3] at h4 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g b = c
⊢ h a = c | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g (f a) = c
⊢ h a = c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g b = c
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_1_5 | [110, 1] | [144, 7] | show h a = c from h4 | case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g (f a) = c
⊢ h a = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
A B C : Type
f : A → B
g : B → C
h : A → C := fun x => g (f x)
a : A
c : C
h1 : ∃ x, (a, x) ∈ graph f ∧ (x, c) ∈ graph g
b : B
h2 : (a, b) ∈ graph f ∧ (b, c) ∈ graph g
h3 : f a = b
h4 : g (f a) = c
⊢ h a = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | assume h1 : one_to_one f | A B C : Type
f : A → B
g : B → C
⊢ one_to_one f → one_to_one g → one_to_one (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
⊢ one_to_one g → one_to_one (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
⊢ one_to_one f → one_to_one g → one_to_one (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | assume h2 : one_to_one g | A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
⊢ one_to_one g → one_to_one (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
h2 : one_to_one g
⊢ one_to_one (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
⊢ one_to_one g → one_to_one (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | define at h1 | A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
h2 : one_to_one g
⊢ one_to_one (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : one_to_one g
⊢ one_to_one (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : one_to_one f
h2 : one_to_one g
⊢ one_to_one (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | define at h2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : one_to_one g
⊢ one_to_one (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ one_to_one (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : one_to_one g
⊢ one_to_one (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | define | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ one_to_one (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ ∀ (x1 x2 : A), (g ∘ f) x1 = (g ∘ f) x2 → x1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ one_to_one (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | fix a1 : A | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ ∀ (x1 x2 : A), (g ∘ f) x1 = (g ∘ f) x2 → x1 = x2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 : A
⊢ ∀ (x2 : A), (g ∘ f) a1 = (g ∘ f) x2 → a1 = x2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
⊢ ∀ (x1 x2 : A), (g ∘ f) x1 = (g ∘ f) x2 → x1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | fix a2 : A | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 : A
⊢ ∀ (x2 : A), (g ∘ f) a1 = (g ∘ f) x2 → a1 = x2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ (g ∘ f) a1 = (g ∘ f) a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 : A
⊢ ∀ (x2 : A), (g ∘ f) a1 = (g ∘ f) x2 → a1 = x2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | define : (g ∘ f) a1 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ (g ∘ f) a1 = (g ∘ f) a2 → a1 = a2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = (g ∘ f) a2 → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ (g ∘ f) a1 = (g ∘ f) a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | define : (g ∘ f) a2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = (g ∘ f) a2 → a1 = a2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = g (f a2) → a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = (g ∘ f) a2 → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | assume h3 : g (f a1) = g (f a2) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = g (f a2) → a1 = a2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
⊢ g (f a1) = g (f a2) → a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | have h4 : f a1 = f a2 := h2 (f a1) (f a2) h3 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
⊢ a1 = a2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
h4 : f a1 = f a2
⊢ a1 = a2 | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_1 | [154, 1] | [168, 7] | show a1 = a2 from h1 a1 a2 h4 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
h4 : f a1 = f a2
⊢ a1 = a2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (x1 x2 : A), f x1 = f x2 → x1 = x2
h2 : ∀ (x1 x2 : B), g x1 = g x2 → x1 = x2
a1 a2 : A
h3 : g (f a1) = g (f a2)
h4 : f a1 = f a2
⊢ a1 = a2
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.comp_def | [170, 1] | [171, 34] | rfl | A B C : Type
g : B → C
f : A → B
x : A
⊢ (g ∘ f) x = g (f x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
g : B → C
f : A → B
x : A
⊢ (g ∘ f) x = g (f x)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | assume h1 : onto f | A B C : Type
f : A → B
g : B → C
⊢ onto f → onto g → onto (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : onto f
⊢ onto g → onto (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
⊢ onto f → onto g → onto (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | assume h2 : onto g | A B C : Type
f : A → B
g : B → C
h1 : onto f
⊢ onto g → onto (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : onto f
h2 : onto g
⊢ onto (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : onto f
⊢ onto g → onto (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | define at h1 | A B C : Type
f : A → B
g : B → C
h1 : onto f
h2 : onto g
⊢ onto (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : onto g
⊢ onto (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : onto f
h2 : onto g
⊢ onto (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | define at h2 | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : onto g
⊢ onto (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ onto (g ∘ f) | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : onto g
⊢ onto (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | define | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ onto (g ∘ f) | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ ∀ (y : C), ∃ x, (g ∘ f) x = y | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ onto (g ∘ f)
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | fix c : C | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ ∀ (y : C), ∃ x, (g ∘ f) x = y | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
⊢ ∃ x, (g ∘ f) x = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
⊢ ∀ (y : C), ∃ x, (g ∘ f) x = y
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | obtain (b : B) (h3 : g b = c) from h2 c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
⊢ ∃ x, (g ∘ f) x = c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
⊢ ∃ x, (g ∘ f) x = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
⊢ ∃ x, (g ∘ f) x = c
TACTIC:
|
https://github.com/djvelleman/HTPILeanPackage.git | 4d23e94fff351c65b5e1345c43451f2aa9908c27 | HTPILib/Chap5.lean | HTPI.Theorem_5_2_5_2 | [173, 1] | [187, 7] | obtain (a : A) (h4 : f a = b) from h1 b | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
⊢ ∃ x, (g ∘ f) x = c | A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
a : A
h4 : f a = b
⊢ ∃ x, (g ∘ f) x = c | Please generate a tactic in lean4 to solve the state.
STATE:
A B C : Type
f : A → B
g : B → C
h1 : ∀ (y : B), ∃ x, f x = y
h2 : ∀ (y : C), ∃ x, g x = y
c : C
b : B
h3 : g b = c
⊢ ∃ x, (g ∘ f) x = c
TACTIC:
|
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