Find a grammar for the language

Question

{a^mbⁿcⁱ | m > n + i}

I've been trying to figure this out for two hours. This is what I have so far.

//To start with as many a's as you want:  
S => a | aA | aS   
//To ensure an a gets added each time a b or c does so there is always at least 1 more a than b's plus c's.  
A => aBb | aaBbCc | aCc   
B => aBb | lambda  
C => ???

I know this is nowhere near correct, which is why I'm asking for help/hints.

Thanks.

Answer 1

Your grammar is not correct.

Read Tips for creating “Context Free Grammar” and linked answers to it, Grammar of your language is

S --> aSc | B    // for every `c` there must be a `a`
B --> aBb | A    // for every `b` there must be a `a`
A --> aA  | a    // generate extra `a`

Answer 2

Build it up from the inside out:

• B → aBb | ε        -- matches aⁱbⁱ
• C → aCc | B       -- matches a^i+jbⁱc^j
• S → aC | aS       -- add at least one more a on the front

This is not minimal -- you could do it with 5 rules and 2 non-terminals.