(** {1 The Patience Solitaire Game} Problem 1 from the {h Verified Software Competition 2014} Patience Solitaire is played by taking cards one-by-one from a deck of cards and arranging them face up in a sequence of stacks arranged from left to right as follows. The very first card from the deck is kept face up to form a singleton stack. Each subsequent card is placed on the leftmost stack where its card value is no greater than the topmost card on that stack. If there is no such stack, then a new stack is started to right of the other stacks. We can do this with positive numbers instead of cards. If the input sequence is 9, 7, 10, 9, 5, 4, and 10, then the stacks develop as {h
```}
<[]>
<[[7, 9]]>
<[[7, 9]], []>
<[[7, 9]], [[9, 10]]>
<[[5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]]>
<[[4, 5, 7, 9]], [[9, 10]], []>
{h ```
} Verify the claim is that the number of stacks at the end of the game is the length of the longest (strictly) increasing subsequence in the input sequence. *) (** {2 Preliminary: pigeon-hole lemma} *) module PigeonHole (** The Why standard library provides a lemma `map.MapInjection.injective_surjective` stating that a map from `(0..n-1)` to `(0..n-1)` that is an injection is also a surjection. This is more or less equivalent to the pigeon-hole lemma. However, we need such a lemma more generally on functions instead of maps. Thus we restate the pigeon-hole lemma here. Proof is left as an exercise. *) use int.Int predicate range (f: int -> int) (n: int) (m:int) = forall i: int. 0 <= i < n -> 0 <= f i < m (** `range f n m` is true when `f` maps the domain `(0..n-1)` into `(0..m-1)` *) predicate injective (f: int -> int) (n: int) (m:int) = forall i j: int. 0 <= i < j < n -> f i <> f j (** `injective f n m` is true when `f` is an injection from `(0..n-1)` to `(0..m-1)` *) (* lemma pigeon_hole2: forall n m:int, f: int -> int. range f n m /\ n > m >= 0 -> not (injective f n m) *) exception Found function shift (f: int -> int) (i:int) : int -> int = fun k -> if k < i then f k else f (k+1) let rec lemma pigeon_hole (n m:int) (f: int -> int) requires { range f n m } requires { n > m >= 0 } variant { m } ensures { not (injective f n m) } = for i = 0 to n-1 do invariant { forall k. 0 <= k < i -> f k <> m-1 } if f i = m-1 then begin (* we have found index i such that f i = m-1 *) for j = i+1 to n-1 do invariant { forall k. i < k < j -> f k <> m-1 } (* we know that f i = f j = m-1 hence we are done *) if f j = m-1 then return done; (* we know that for all k <> i, f k <> m-1 *) let g = shift f i in assert { range g (n-1) (m-1) }; pigeon_hole (n-1) (m-1) g; return end done; (* we know that for all k, f k <> m-1 *) assert { range f n (m-1) }; pigeon_hole n (m-1) f end (** {2 Patience idiomatic code} *) module PatienceCode use int.Int use list.List use list.RevAppend (** this code was the one written initially, without any specification, except for termination, ans unreachability of the 'absurd' branch'. It can be tested, see below. *) type card = int (** stacks are well-formed if they are non-empty *) predicate wf_stacks (stacks: list (list card)) = match stacks with | Nil -> true | Cons Nil _ -> false | Cons (Cons _ _) rem -> wf_stacks rem end (** concatenation of well-formed stacks is well-formed *) let rec lemma wf_rev_append_stacks (s1 s2: list (list int)) requires { wf_stacks s1 } requires { wf_stacks s2 } variant { s1 } ensures { wf_stacks (rev_append s1 s2) } = match s1 with | Nil -> () | Cons Nil _ -> absurd | Cons s rem -> wf_rev_append_stacks rem (Cons s s2) end (** `push_card c stacks acc` pushes card `c` on stacks `stacks`, assuming `acc` is an accumulator (in reverse order) of stacks where `c` could not be pushed. *) let rec push_card (c:card) (stacks : list (list card)) (acc : list (list card)) : list (list card) requires { wf_stacks stacks } requires { wf_stacks acc } variant { stacks } ensures { wf_stacks result } = match stacks with | Nil -> (* we put card `c` in a new stack *) rev_append (Cons (Cons c Nil) acc) Nil | Cons stack remaining_stacks -> match stack with | Nil -> absurd (* because `wf_stacks stacks` *) | Cons c' _ -> if c <= c' then (* card is placed on the leftmost stack where its card value is no greater than the topmost card on that stack *) rev_append (Cons (Cons c stack) acc) remaining_stacks else (* try next stack *) push_card c remaining_stacks (Cons stack acc) end end let rec play_cards (input: list card) (stacks: list (list card)) : list (list card) requires { wf_stacks stacks } variant { input } ensures { wf_stacks result } = match input with | Nil -> stacks | Cons c rem -> let stacks' = push_card c stacks Nil in play_cards rem stacks' end let play_game (input: list card) : list (list card) = play_cards input Nil (** test, can be run using `why3 patience.mlw --exec PatienceCode.test` *) let test () = (** the list given in the problem description 9, 7, 10, 9, 5, 4, and 10 *) play_game (Cons 9 (Cons 7 (Cons 10 (Cons 9 (Cons 5 (Cons 4 (Cons 10 Nil))))))) end (** {2 Abstract version of Patience game} *) module PatienceAbstract use int.Int (** To specify the expected property of the Patience game, we first provide an abstract version, working on a abstract state that includes a lot of information regarding the positions of the cards in the stack and so on. This abstract state should then be including in the real code as a ghost state, with a gluing invariant that matches the ghost state and the concrete stacks of cards. *) type card = int (** {3 Abstract state} *) use map.Map use map.Const type state = { ghost mutable num_stacks : int; (** number of stacks built so far *) ghost mutable num_elts : int; (** number of cards already seen *) ghost mutable values : map int card; (** cards values seen, indexed in the order they have been seen, from `0` to `num_elts-1` *) ghost mutable stack_sizes : map int int; (** sizes of these stacks, numbered from `0` to `num_stacks - 1` *) ghost mutable stacks : map int (map int int); (** indexes of the cards in respective stacks *) ghost mutable positions : map int (int,int); (** table that given a card index, provides its position, i.e. in which stack it is and at which height *) ghost mutable preds : map int int; (** predecessors of cards, i.e. for each card index `i`, `preds[i]` provides an index of a card in the stack on the immediate left, whose value is smaller. Defaults to `-1` if the card is on the leftmost stack. *) } (** {3 Invariants on the abstract state} *) predicate inv (s:state) = 0 <= s.num_stacks <= s.num_elts (** the number of stacks is less or equal the number of cards *) /\ (s.num_elts > 0 -> s.num_stacks > 0) (** when there is at least one card, there is at least one stack *) /\ (forall i. 0 <= i < s.num_stacks -> s.stack_sizes[i] >= 1 (** stacks are non-empty *) /\ forall j. 0 <= j < s.stack_sizes[i] -> 0 <= s.stacks[i][j] < s.num_elts) (** contents of stacks are valid card indexes *) /\ (forall i. 0 <= i < s.num_elts -> let is,ip = s.positions[i] in 0 <= is < s.num_stacks && let st = s.stacks[is] in 0 <= ip < s.stack_sizes[is] && st[ip] = i) (** the position table of cards is correct, i.e. when `(is,ip) = s.positions[i]` then card `i` indeed occurs in stack `is` at height `ip` *) /\ (forall is. 0 <= is < s.num_stacks -> forall ip. 0 <= ip < s.stack_sizes[is] -> let idx = s.stacks[is][ip] in (is,ip) = s.positions[idx]) (** positions is the proper inverse of stacks *) /\ (forall i. 0 <= i < s.num_stacks -> let stack_i = s.stacks[i] in forall j,k. 0 <= j < k < s.stack_sizes[i] -> stack_i[j] < stack_i[k]) (** in a given stack, indexes are increasing from bottom to top *) /\ (forall i. 0 <= i < s.num_stacks -> let stack_i = s.stacks[i] in forall j,k. 0 <= j <= k < s.stack_sizes[i] -> s.values[stack_i[j]] >= s.values[stack_i[k]]) (** in a given stack, card values are decreasing from bottom to top *) /\ (forall i. 0 <= i < s.num_elts -> let pred = s.preds[i] in -1 <= pred < s.num_elts && (** the predecessor is a valid index or `-1` *) pred < i /\ (** predecessor is always a smaller index *) let is,_ip = s.positions[i] in if pred < 0 then is = 0 (** if predecessor is `-1` then `i` is in leftmost stack *) else s.values[pred] < s.values[i] /\ (** if predecessor is not `-1`, it denotes a card with smaller value... *) is > 0 && (** ...the card is not on the leftmost stack... *) let ps,_pp = s.positions[pred] in ps = is - 1) (** ...and predecessor is in the stack on the immediate left *) (** {2 Programs} *) use ref.Ref exception Return int (** `play_card c i s` pushes the card `c` on state `s` *) let ghost play_card (c:card) (s:state) : unit requires { inv s } writes { s } ensures { inv s } ensures { s.num_elts = (old s).num_elts + 1 } ensures { s.values = (old s).values[(old s).num_elts <- c] } = let ghost pred = ref (-1) in try for i = 0 to s.num_stacks - 1 do invariant { if i=0 then !pred = -1 else let stack_im1 = s.stacks[i-1] in let stack_im1_size = s.stack_sizes[i-1] in let top_stack_im1 = stack_im1[stack_im1_size - 1] in !pred = top_stack_im1 /\ c > s.values[!pred] /\ 0 <= !pred < s.num_elts /\ let ps,_pp = s.positions[!pred] in ps = i - 1 } let stack_i = s.stacks[i] in let stack_i_size = s.stack_sizes[i] in let top_stack_i = stack_i[stack_i_size - 1] in if c <= s.values[top_stack_i] then raise (Return i); assert { 0 <= top_stack_i < s.num_elts }; assert { let is,ip = s.positions[top_stack_i] in 0 <= is < s.num_stacks && 0 <= ip < s.stack_sizes[is] && s.stacks[is][ip] = top_stack_i && is = i /\ ip = stack_i_size - 1 }; pred := top_stack_i done; (* we add a new stack *) let idx = s.num_elts in let i = s.num_stacks in let stack_i = s.stacks[i] in let new_stack_i = stack_i[0 <- idx] in s.num_elts <- idx + 1; s.values <- s.values[idx <- c]; s.num_stacks <- s.num_stacks + 1; s.stack_sizes <- s.stack_sizes[i <- 1]; s.stacks <- s.stacks[i <- new_stack_i]; s.positions <- s.positions[idx <- i,0]; s.preds <- s.preds[idx <- !pred] with Return i -> let stack_i = s.stacks[i] in let stack_i_size = s.stack_sizes[i] in (* we put c on top of stack i *) let idx = s.num_elts in let new_stack_i = stack_i[stack_i_size <- idx] in s.num_elts <- idx + 1; s.values <- s.values[idx <- c]; (* s.num_stacks unchanged *) s.stack_sizes <- s.stack_sizes[i <- stack_i_size + 1]; s.stacks <- s.stacks[i <- new_stack_i]; s.positions <- s.positions[idx <- i,stack_i_size]; s.preds <- s.preds[idx <- !pred]; end use list.List use list.Length use list.NthNoOpt let rec play_cards (input: list int) (s: state) : unit requires { inv s } variant { input } writes { s } ensures { inv s } ensures { s.num_elts = (old s).num_elts + length input } ensures { forall i. 0 <= i < (old s).num_elts -> s.values[i] = (old s).values[i] } ensures { forall i. (old s).num_elts <= i < s.num_elts -> s.values[i] = nth (i - (old s).num_elts) input } = match input with | Nil -> () | Cons c rem -> play_card c s; play_cards rem s end type seq 'a = { seqlen: int; seqval: map int 'a } predicate increasing_subsequence (s:seq int) (l:list int) = 0 <= s.seqlen <= length l && (* subsequence *) ((forall i. 0 <= i < s.seqlen -> 0 <= s.seqval[i] < length l) /\ (forall i,j. 0 <= i < j < s.seqlen -> s.seqval[i] < s.seqval[j])) (* increasing *) && (forall i,j. 0 <= i < j < s.seqlen -> nth s.seqval[i] l < nth s.seqval[j] l) use PigeonHole let ghost play_game (input: list int) : state ensures { exists s: seq int. s.seqlen = result.num_stacks /\ increasing_subsequence s input } ensures { forall s: seq int. increasing_subsequence s input -> s.seqlen <= result.num_stacks } = let s = { num_elts = 0; values = Const.const (-1) ; num_stacks = 0; stack_sizes = Const.const 0; stacks = Const.const (Const.const (-1)); positions = Const.const (-1,-1); preds = Const.const (-1); } in play_cards input s; (** This is ghost code to build an increasing subsequence of maximal length *) let ns = s.num_stacks in if ns = 0 then begin assert { input = Nil }; let seq = { seqlen = 0 ; seqval = Const.const (-1) } in assert { increasing_subsequence seq input }; s end else let last_stack = s.stacks[ns-1] in let idx = ref (last_stack[s.stack_sizes[ns-1]-1]) in let seq = ref (Const.const (-1)) in for i = ns-1 downto 0 do invariant { -1 <= !idx < s.num_elts } invariant { i >= 0 -> !idx >= 0 && let is,_ = s.positions[!idx] in is = i } invariant { i+1 < ns -> !idx < !seq[i+1] } invariant { 0 <= i < ns-1 -> s.values[!idx] < s.values[!seq[i+1]] } invariant { forall j. i < j < ns -> 0 <= !seq[j] < s.num_elts } invariant { forall j,k. i < j < k < ns -> !seq[j] < !seq[k] } invariant { forall j,k. i < j < k < ns -> s.values[!seq[j]] < s.values[!seq[k]] } seq := !seq[i <- !idx]; idx := s.preds[!idx]; done; let sigma = { seqlen = ns ; seqval = !seq } in assert { forall i. 0 <= i < length input -> nth i input = s.values[i] }; assert { increasing_subsequence sigma input }; (** These are assertions to prove that no increasing subsequence of length larger than the number of stacks may exists *) assert { (* non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks -> let f = fun i -> let si = sigma.seqval[i] in let stack_i,_ = s.positions[si] in stack_i in range f sigma.seqlen s.num_stacks && not (injective f sigma.seqlen s.num_stacks) }; assert { (* non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks -> exists i,j. 0 <= i < j < sigma.seqlen && let si = sigma.seqval[i] in let sj = sigma.seqval[j] in let stack_i,_pi = s.positions[si] in let stack_j,_pj = s.positions[sj] in stack_i = stack_j }; assert { (* contradiction from non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > s.num_stacks -> forall i,j. 0 <= i < j < sigma.seqlen -> let si = sigma.seqval[i] in let sj = sigma.seqval[j] in let stack_i,pi = s.positions[si] in let stack_j,pj = s.positions[sj] in stack_i = stack_j -> si < sj && pi < pj && s.values[si] < s.values[sj] }; s let ghost test () = (* the list given in the problem description 9, 7, 10, 9, 5, 4, and 10 *) play_game (Cons 9 (Cons 7 (Cons 10 (Cons 9 (Cons 5 (Cons 4 (Cons 10 Nil))))))) end (** {2 Gluing abstract version with the original idiomatic code} *) module PatienceFull use int.Int use PatienceAbstract (** glue between the ghost state and the stacks of cards *) use list.List use list.Length use list.NthNoOpt use map.Map predicate glue_stack (s:state) (i:int) (st:list card) = length st = s.stack_sizes[i] /\ let stack_i = s.stacks[i] in forall j. 0 <= i < length st -> nth j st = s.values[stack_i[j]] predicate glue (s:state) (st:list (list card)) = length st = s.num_stacks /\ forall i. 0 <= i < length st -> glue_stack s i (nth i st) (** {3 playing a card} *) use list.RevAppend use ref.Ref exception Return (*** FIXME: not proved let play_card (c:card) (old_stacks : list (list card)) (ghost state:state) : list (list card) requires { inv state } requires { glue state old_stacks } writes { state } ensures { inv state } ensures { state.num_elts = (old state).num_elts + 1 } ensures { state.values = (old state).values[(old state).num_elts <- c] } ensures { glue state result } = let acc = ref Nil in let rem_stacks = ref old_stacks in let ghost pred = ref (-1) in let ghost i = ref 0 in try while !rem_stacks <> Nil do invariant { 0 <= !i <= state.num_stacks } invariant { if !i = 0 then !pred = -1 else let stack_im1 = state.stacks[!i-1] in let stack_im1_size = state.stack_sizes[!i-1] in let top_stack_im1 = stack_im1[stack_im1_size - 1] in !pred = top_stack_im1 /\ c > state.values[!pred] /\ 0 <= !pred < state.num_elts /\ let ps,_pp = state.positions[!pred] in ps = !i - 1 } invariant { old_stacks = rev_append !acc !rem_stacks } invariant { forall j. 0 <= j < !i -> glue_stack state j (nth (!i - j) !acc) } invariant { forall j. !i <= j < state.num_stacks -> glue_stack state j (nth (j - !i) !rem_stacks) } variant { !rem_stacks } match !rem_stacks with | Nil -> absurd | Cons stack remaining_stacks -> rem_stacks := remaining_stacks; match stack with | Nil -> assert { glue_stack state !i stack }; absurd | Cons c' _ -> if c <= c' then begin acc := Cons (Cons c stack) !acc; raise Return; end; let ghost stack_i = state.stacks[!i] in let ghost stack_i_size = state.stack_sizes[!i] in let ghost top_stack_i = stack_i[stack_i_size - 1] in assert { 0 <= top_stack_i < state.num_elts }; assert { let is,ip = state.positions[top_stack_i] in 0 <= is < state.num_stacks && 0 <= ip < state.stack_sizes[is] && state.stacks[is][ip] = top_stack_i && is = !i /\ ip = stack_i_size - 1 }; i := !i + 1; acc := Cons stack !acc; pred := top_stack_i end end done; (* we add a new stack *) let ghost idx = state.num_elts in let ghost i = state.num_stacks in let ghost stack_i = state.stacks[i] in let ghost new_stack_i = stack_i[0 <- idx] in state.num_elts <- idx + 1; state.values <- state.values[idx <- c]; state.num_stacks <- state.num_stacks + 1; state.stack_sizes <- state.stack_sizes[i <- 1]; state.stacks <- state.stacks[i <- new_stack_i]; state.positions <- state.positions[idx <- i,0]; state.preds <- state.preds[idx <- !pred]; (* we put card [c] in a new stack *) rev_append (Cons (Cons c Nil) !acc) Nil with Return -> let ghost stack_i = state.stacks[!i] in let ghost stack_i_size = state.stack_sizes[!i] in let ghost top_stack_i = stack_i[stack_i_size - 1] in assert { c <= state.values[top_stack_i] }; (* we put c on top of stack i *) let ghost idx = state.num_elts in let ghost new_stack_i = stack_i[stack_i_size <- idx] in state.num_elts <- idx + 1; state.values <- state.values[idx <- c]; (* state.num_stacks unchanged *) state.stack_sizes <- state.stack_sizes[!i <- stack_i_size + 1]; state.stacks <- state.stacks[!i <- new_stack_i]; state.positions <- state.positions[idx <- !i,stack_i_size]; state.preds <- state.preds[idx <- !pred]; (* card is placed on the leftmost stack where its card value is no greater than the topmost card on that stack *) rev_append !acc !rem_stacks end *) (*** a version closer to the original code let play_card (c:card) (old_stacks : list (list card)) (ghost state:state) : list (list card) requires { inv state } requires { glue state old_stacks } writes { state } ensures { inv state } ensures { state.num_elts = (old state).num_elts + 1 } ensures { state.values = (old state).values[(old state).num_elts <- c] } ensures { glue state result } = let i = ref 0 in let pred = ref (-1) in let rec push_card (c:card) (st : list (list card)) (acc : list (list card)) : list (list card) requires { old_stacks = rev_append acc st } variant { st } = match st with | Nil -> (* we put card [c] in a new stack *) rev_append (Cons (Cons c Nil) acc) Nil | Cons stack remaining_stacks -> match stack with | Nil -> assert { glue_stack state !i stack }; absurd | Cons c' _ -> if c <= c' then (* card is placed on the leftmost stack where its card value is no greater than the topmost card on that stack *) rev_append (Cons (Cons c stack) acc) remaining_stacks else (* try next stack *) push_card c remaining_stacks (Cons stack acc) end end in let new_stacks = push_card c old_stacks Nil in let idx = state.num_elts in state.num_elts <- idx + 1; state.values <- state.values[idx <- c]; new_stacks *) (*** {3 playing cards} *) (*** let rec play_cards (input: list card) (stacks: list (list card)) (ghost state:state) : list (list card) requires { inv state } requires { glue state stacks } variant { input } (* writes { state } *) ensures { inv state } ensures { state.num_elts = (old state).num_elts + length input } ensures { forall i. 0 <= i < (old state).num_elts -> state.values[i] = (old state).values[i] } ensures { forall i. (old state).num_elts <= i < state.num_elts -> state.values[i] = nth (i - (old state).num_elts) input } ensures { glue state result } = match input with | Nil -> stacks | Cons c rem -> let stacks' = play_card c stacks state in play_cards rem stacks' state end *) (*** {3 playing a whole game} *) (*** type seq 'a = { seqlen: int; seqval: map int 'a } (** a sequence is defined by a length and a mapping *) (** definition of an increasing sub-sequence of a list of card *) predicate increasing_subsequence (sigma:seq int) (l:list card) = 0 <= sigma.seqlen <= length l (** the length of [sigma] is at most the number of cards *) && (forall i. 0 <= i < sigma.seqlen -> 0 <= sigma.seqval[i] < length l) (** [sigma] maps indexes to valid indexes in the card list *) && (forall i,j. 0 <= i < j < sigma.seqlen -> sigma.seqval[i] < sigma.seqval[j]) (** [sigma] is an increasing sequence of indexes *) && (forall i,j. 0 <= i < j < sigma.seqlen -> nth sigma.seqval[i] l < nth sigma.seqval[j] l) (** the card values denoted by [sigma] are increasing *) use PigeonHole let play_game (input: list card) : list (list card) requires { length input > 0 } ensures { exists sigma: seq int. sigma.seqlen = length result /\ increasing_subsequence sigma input } ensures { forall sigma: seq int. increasing_subsequence sigma input -> sigma.seqlen <= length result } = let ghost state = { num_elts = 0; values = Const.const (-1) ; num_stacks = 0; stack_sizes = Const.const 0; stacks = Const.const (Const.const (-1)); positions = Const.const (-1,-1); preds = Const.const (-1); } in let final_stacks = play_cards input Nil state in assert { forall i. 0 <= i < length input -> nth i input = state.values[i] }; (** This is ghost code to build an increasing subsequence of maximal length *) let ghost ns = state.num_stacks in let ghost _sigma = if ns = 0 then begin assert { input = Nil }; absurd (* TODO: if input is empty, we may be able to prove that: let sigma = { seqlen = 0 ; seqval = Const.const (-1) } in assert { increasing_subsequence sigma input }; sigma *) end else let ghost last_stack = state.stacks[ns-1] in let ghost idx = ref (last_stack[state.stack_sizes[ns-1]-1]) in let ghost seq = ref (Const.const (-1)) in for i = ns-1 downto 0 do invariant { -1 <= !idx < state.num_elts } invariant { i >= 0 -> !idx >= 0 && let is,_ = state.positions[!idx] in is = i } invariant { i+1 < ns -> !idx < !seq[i+1] } invariant { 0 <= i < ns-1 -> state.values[!idx] < state.values[!seq[i+1]] } invariant { forall j. i < j < ns -> 0 <= !seq[j] < state.num_elts } invariant { forall j,k. i < j < k < ns -> !seq[j] < !seq[k] } invariant { forall j,k. i < j < k < ns -> state.values[!seq[j]] < state.values[!seq[k]] } seq := !seq[i <- !idx]; idx := state.preds[!idx]; done; let ghost sigma = { seqlen = ns ; seqval = !seq } in assert { increasing_subsequence sigma input }; (** These are assertions to prove that no increasing subsequence of length larger than the number of stacks may exists *) assert { (* non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks -> let f = fun i -> let si = sigma.seqval[i] in let stack_i,_ = state.positions[si] in stack_i in range f sigma.seqlen state.num_stacks && not (injective f sigma.seqlen state.num_stacks) }; assert { (* non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks -> exists i,j. 0 <= i < j < sigma.seqlen && let si = sigma.seqval[i] in let sj = sigma.seqval[j] in let stack_i,_pi = state.positions[si] in let stack_j,_pj = state.positions[sj] in stack_i = stack_j }; assert { (* contradiction from non-injectivity *) forall sigma: seq int. increasing_subsequence sigma input /\ sigma.seqlen > state.num_stacks -> forall i,j. 0 <= i < j < sigma.seqlen -> let si = sigma.seqval[i] in let sj = sigma.seqval[j] in let stack_i,pi = state.positions[si] in let stack_j,pj = state.positions[sj] in stack_i = stack_j -> si < sj && pi < pj && state.values[si] < state.values[sj] }; sigma in final_stacks *) end