% Convention: Hi are hypotheses, Ci are conclusions. % Everything after a % is a comment. % You are not required to comment your tutch proofs. % Annotated version of proof in Tutch manual. proof ExNotImpNotAll : (?x:t. ~A(x)) => ~!x:t. A(x) = begin [ ?x:t. ~A(x); % H1 [ !x:t. A(x); % H2 [ c: t, ~A(c); % H3 A(c); % C1: !E on H2 and c (H3) F ]; % C2: =>E on H3 and C1 F ]; % C3: ?E on H1 and H3..C2 ~!x:t. A(x) ]; % C4: ~I on H2..C3 (?x:t. ~A(x)) => ~!x:t. A(x); % C5: =>I on H1..C4 end; proof swap : (?y:t.!x:t.A(x,y)) => (!x:t.?y:t.A(x,y)) = begin [ ?y:t.!x:t.A(x,y); % H1 [ c : t, !x : t.A(x, c); % H2 [ d : t; % H3 A(d, c); % C1: !E on H2 and d (H3) ?y:t.A(d, y) ]; % C2: ?I on C1 and c !x:t.?y:t.A(x,y) ]; % C3: !I on C2 and d !x:t.?y:t.A(x,y) ]; % C4: ?E on H1 and H2..C3 (?y:t.!x:t.A(x,y)) => (!x:t.?y:t.A(x,y)); % C5: =>I on H1..C4 end;