let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty finite-yielding MSAlgebra over S
for V being Variables of A
for X being SetWithCompoundTerm of S,V
for G being non empty non void Circuit-like ManySortedSign
for C being non-empty Circuit of G st C calculates X,A holds
for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let A be non-empty finite-yielding MSAlgebra over S; :: thesis: for V being Variables of A
for X being SetWithCompoundTerm of S,V
for G being non empty non void Circuit-like ManySortedSign
for C being non-empty Circuit of G st C calculates X,A holds
for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let V be Variables of A; :: thesis: for X being SetWithCompoundTerm of S,V
for G being non empty non void Circuit-like ManySortedSign
for C being non-empty Circuit of G st C calculates X,A holds
for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let X be SetWithCompoundTerm of S,V; :: thesis: for G being non empty non void Circuit-like ManySortedSign
for C being non-empty Circuit of G st C calculates X,A holds
for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let G be non empty non void Circuit-like ManySortedSign ; :: thesis: for C being non-empty Circuit of G st C calculates X,A holds
for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let C be non-empty Circuit of G; :: thesis: ( C calculates X,A implies for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) ) )
assume A1: C calculates X,A ; :: thesis: for f being SortMap of X,A,C
for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let f be SortMap of X,A,C; :: thesis: for t being Term of S,V st t in Subtrees X holds
for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
consider g being Function such that
A2: f,g form_embedding_of X -Circuit A,C by A1, Def17;
A3: f preserves_inputs_of X -CircuitStr ,G by A1, Def17;
A4: f,g form_morphism_between X -CircuitStr ,G by A2;
let t be Term of S,V; :: thesis: ( t in Subtrees X implies for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) ) )
assume A5: t in Subtrees X ; :: thesis: for s being State of C holds
( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
let s be State of C; :: thesis: ( Following (s,(1 + (height (dom t)))) is_stable_at f . t & ( for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) ) )
reconsider s9 = s * f as State of (X -Circuit A) by A2, Th44;
reconsider t9 = t as Vertex of (X -CircuitStr) by A5;
A6: Following (s9,(1 + (height (dom t)))) is_stable_at t9 by Th21;
A7: Following (s9,(1 + (height (dom t)))) = (Following (s,(1 + (height (dom t))))) * f by A2, A3, Th47;
hence Following (s,(1 + (height (dom t)))) is_stable_at f . t by A2, A3, A6, Th49; :: thesis: for s9 being State of (X -Circuit A) st s9 = s * f holds
for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A)
let s9 be State of (X -Circuit A); :: thesis: ( s9 = s * f implies for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) )
assume A8: s9 = s * f ; :: thesis: for h being CompatibleValuation of s9 holds (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A)
let h be CompatibleValuation of s9; :: thesis: (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A)
A9: dom f = the carrier of (X -CircuitStr) by A4;
(Following (s9,(1 + (height (dom t))))) . t9 = t @ (h,A) by Th21;
hence (Following (s,(1 + (height (dom t))))) . (f . t) = t @ (h,A) by A7, A8, A9, FUNCT_1:13; :: thesis: verum