hcju/isabelleps huggingface dataset:
add_sheaf_spec_in_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set"   and t :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit"   and "s \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"   and "t \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"   and "U \<subseteq> comm_ring.spectrum R addition multiplication zero unit" shows "comm_ring.add_sheaf_spec R addition multiplication zero U s t \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
complete_lattice_ccpo': fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.ccpo Sup less_eq (mk_less less_eq)"
power_mono_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "less_eq (power.power one times a n) (power.power one times b n) = less_eq a b"
strip_list_comb_const: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and name :: "name"   and xs :: "'b list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app (const name) xs) = (const name, xs)"
fold_Option_bind_eq_Some_start_not_None: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a option"   and list :: "'b list"   and start :: "'a option"   and res :: "'a" assumes "fold (\<lambda>new option. Option.bind option (f new)) list start = Some res" shows "start \<noteq> None"
bind_left_fail_SE: fixes P :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'a) option" shows "bind\<^sub>S\<^sub>E fail\<^sub>S\<^sub>E P = fail\<^sub>S\<^sub>E"
Ex_finite_integrable_function: shows "\<exists>h\<in>borel_measurable P. integral\<^sup>N P h \<noteq> \<infinity> \<and> (\<forall>x\<in>space P. 0 < h x \<and> h x < \<infinity>)"
arr_Set_incl_SetI: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "arr_Set \<alpha> (incl_Rel A B)"
e4: shows "pdevs_val e (fst (trunc_bound_pdevs p (add_pdevs (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst Y)) (snd (fst X))))) (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst X)) (snd (fst Y)))))))) = pdevs_val e (add_pdevs (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst Y)) (snd (fst X))))) (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst X)) (snd (fst Y)))))) + e4"
lexordp_eq_simps_3: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "ord.lexordp_eq less (x # xs) [] = False"
MATCH1_mono: fixes P :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option"   and P' :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and \<sigma>' :: "'e \<Rightarrow> ('d, 'f) Term.term option"   and M :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset" assumes "MATCH1 (P, \<sigma>) (P', \<sigma>')" shows "MATCH1 (P + M, \<sigma>) (P' + M, \<sigma>')"
arr_Rel_ArrDom_in_Vset: shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
NatTransComp_defs_3: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "MakeNT \<eta> \<equiv> \<lparr>NTDom = NTDom \<eta>, NTCod = NTCod \<eta>, NatTransMap = restrict (($$) \<eta>) (obj\<^bsub>NTCatDom \<eta>\<^esub>)\<rparr>"
oc_projection_alt_def: fixes W :: "'a vec set"   and x :: "'a vec"   and y1 :: "'a vec"   and y2 :: "'a vec" assumes "W \<subseteq> carrier_vec n"   and "x \<in> carrier_vec n"   and "y1 \<in> W"   and "x - y1 \<in> orthogonal_complement W"   and "y2 \<in> W"   and "x - y2 \<in> orthogonal_complement W" shows "y1 = y2"
cut0: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "{} \<turnstile> A"   and "insert A H \<turnstile> B" shows "H \<turnstile> B"
ffield_vempty: shows "\<F>\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
that: fixes f :: "nat \<Rightarrow> 'a" assumes "A = range f"   and "inj f" shows "thesis"
neighbors_def: fixes g :: "graph"   and v :: "nat" shows "neighbors g v \<equiv> map (\<lambda>f. f \<bullet> v) (facesAt g v)"
PlaneGraphsP_def: fixes p :: "nat" shows "PlaneGraphs\<^bsub>p\<^esub> \<equiv> {g. Seed\<^bsub>p\<^esub> [next_plane\<^bsub>p\<^esub>]\<rightarrow>* g \<and> final g}"
orthogonal_subspace_decomp_exists: fixes S :: "'a set"   and x :: "'a" obtains y :: "'a" and z :: "'a" where "y \<in> span S" and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w" and "x = y + z"
characteristic_weak_formula_is_weak: fixes P :: "'state" shows "S_transform.weak_formula (S_transform.characteristic_weak_formula P)"
abs_eq: fixes x :: "'c list" assumes "eq_onp (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None) x x" shows "member (Abs_dlist x) = equal_base.list_member ceq' x"
endomorph_has_eigenvector: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "FinDimVectorSpace smult V"   and "\<And>p. 0 < degree p \<Longrightarrow> \<exists>c. poly p c = (0::'a)"   and "0 < R_scalar_mult.dim_R UNIV smult V"   and "VectorSpaceEnd smult V T" shows "\<exists>c u. u \<in> V \<and> u \<noteq> (0::'b) \<and> T u = smult c u"
simps_1: fixes n :: "int"   and a :: "'av"   and S :: "'av Abs_State.st option" shows "afilter (N n) a S = (if test_num' n a then S else None)"
ev_wait_all_weakly_step_consistent: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and tid :: "thread_id_t"   and ipt :: "int_point_t" assumes "vpeq u s1 s2"   and "vpeq (Step.partition tid) s1 s2"   and "atomic_step_invariant s1"   and "atomic_step_invariant s2"   and "atomic_step_precondition s1 (current s1) ipt"   and "atomic_step_precondition s1 (current s1) ipt" shows "vpeq u (atomic_step_ev_wait_all tid s1) (atomic_step_ev_wait_all tid s2)"
\<mu>_stable_suffix: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" assumes "\<mu>_stable \<phi> w" shows "\<mu>_stable \<phi> (suffix i w)"
elims_2: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b negation_type list list" assumes "dnf_to_bool x xa" obtains f :: "'b \<Rightarrow> bool" and as :: "'b negation_type list" and ass :: "'b negation_type list list" where "x = f" and "xa = as # ass" and "cnf_to_bool f as \<or> dnf_to_bool f ass"
div0_uint64_def: fixes x :: "uint64" shows "div0_uint64 x = undefined (div) x 0"
Trg_Dom_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Trg (Dom t) = Trg t \<and> Trg (Cod t) = Trg t"
Rm_2: shows "LINT x|M. fm x * real_transfer_operator g x = LINT x|M. fm (T x) * g x"
indep_vars_restrict: fixes M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'a \<Rightarrow> 'e"   and I :: "'d set"   and L :: "'f set"   and K :: "'f \<Rightarrow> 'd set" assumes "indep_vars M' X I"   and "\<And>j. j \<in> L \<Longrightarrow> K j \<subseteq> I"   and "disjoint_family_on K L" shows "indep_vars (\<lambda>j. Pi\<^sub>M (K j) M') (\<lambda>j \<omega>. \<lambda>i\<in>K j. X i \<omega>) L"
backdoor_def: shows "backdoor \<equiv> \<lambda>\<psi>. \<^bold>\<lambda>x. \<lparr>\<^bold>\<lambda>x. \<^bold>\<forall>p. p \<^bold>\<rightarrow> p,\<^bold>\<iota>y. y \<^bold>= x \<^bold>& \<psi> (x\<^sup>P)\<rparr>"
source_zero_lens: shows "\<S>\<^bsub>0\<^sub>L\<^esub> = UNIV"
fps_fps_exp_compose_minus: fixes c :: "'a" shows "fps_exp c oo - fps_X = fps_exp (- c)"
semiring_ow_axioms: shows "semiring_ow U (+\<^sub>o\<^sub>w) (*\<^sub>o\<^sub>w)"
f_Exec_Stream_Acc_Output_Init_eq_f_Exec_Stream_Acc_Output_Cons: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun xs c = output_fun c # f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c"
fls_minus_nth: fixes f :: "'a fls"   and g :: "'a fls"   and n :: "int" shows "fls_nth (f - g) n = fls_nth f n - fls_nth g n"
ABind_nonrec_def: fixes x :: "Vars.var"   and e :: "exp" shows "ABind_nonrec x e = (\<Lambda> i. if isVal e \<or> x--x\<notin>snd i then fst i x else up\<cdot>0)"
one_dim_onorm': fixes f :: "'a \<Rightarrow> 'b" assumes "clinear f" shows "onorm f = cmod (one_dim_iso (f (1::'a)))"
replicate_eqI: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs = n"   and "\<And>y. y \<in> set xs \<Longrightarrow> y = x" shows "xs = replicate n x"
is_extensional: fixes f :: "'b" assumes "\<not> F.B.arr f" shows "G f = G.B.null"
characteristic_formula_supp_aux: fixes P :: "'state" shows "supp (Abs_bset {distinguishing_formula P Q |Q. \<not> P =\<cdot> Q}) \<subseteq> supp P"
assms_1: shows "constructive_security_aux_obsf real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 adv1"
blin_matrix_vector_mult: fixes A :: "(('b, 'c) vec, 'd) vec" shows "bounded_linear ((*v) A)"
card_eq_sum: fixes A :: "'k set" shows "card A = (\<Sum>x\<in>A. 1)"
less_le_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "strict_prefix x y"   and "prefix y z" shows "strict_prefix x z"
sup_least: fixes y :: "'a multiset"   and x :: "'a multiset"   and z :: "'a multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
simps_2: fixes n :: "nat" shows "num_params (poly.Bound n) = Suc n"
that: fixes as' :: "'edge list" assumes "m -as'\<rightarrow>\<^sub>\<iota>* m'"   and "preds (slice_kinds S as') (cf # cfs)"   and "slice_edges S cs as' = []" shows "thesis"
cos_float_interval: fixes x :: "float interval"   and p :: "nat" shows "cos ` set_of (real_interval x) \<subseteq> set_of (real_interval (cos_float_interval p x))"
subnet_of_int_def: fixes x :: "int \<times> int" shows "subnet_of x \<equiv> {{(a, b). a = fst x}}"
closed_vimage_fst: fixes S :: "'b set" assumes "closed S" shows "closed (fst -` S)"
simps_1: fixes r :: "'rule" shows "inPorts (Rule r) = f_antecedent r"
measurable_emeasure_kernel: fixes K :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and A :: "'d set" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "A \<in> sets N" shows "(\<lambda>a. emeasure (K a) A) \<in> borel_measurable M"
distinct_entries: fixes t :: "('b, 'c) RBT.rbt" shows "distinct (map fst (RBT.entries t))"
lookup_pair_tl': fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.lookup_pair comp (tl xs) k = (if k = fst (hd xs) then 0::'b else comparator.lookup_pair comp xs k)"
cong_hcomp2: fixes P :: "conc_interface"   and Q :: "conc_interface"   and P' :: "conc_interface" assumes "P \<simeq> Q" shows "P \<otimes>\<^sub>c P' \<simeq> Q \<otimes>\<^sub>c P'"
elims_3: fixes x :: "atom" assumes "\<not> is_Less x" obtains v :: "nat" and va :: "nat" where "x = Eq v va"
intros_1: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A ensures B" shows "(A, B) \<in> leads F"
fresh_FDERIV_floatarith: fixes ode_e :: "floatarith"   and i :: "nat"   and ds :: "floatarith list" assumes "fresh_floatarith ode_e i"   and "fresh_floatariths ds i"   and "length ds = DIM('b)" shows "fresh_floatarith (FDERIV_floatarith ode_e [0..<DIM('b)] ds) i"
sorted_induct_remove1: fixes cmp :: "'b comparator"   and xs :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "Sorting_Algorithms.sorted cmp xs"   and "P []"   and "\<And>x xs. \<lbrakk>Sorting_Algorithms.sorted cmp xs; P (remove1 x xs); x \<in> set xs; hd (filter (\<lambda>y. compare cmp x y = Equiv) xs) = x; \<And>y. y \<in> set xs \<Longrightarrow> compare cmp x y \<noteq> Greater\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
snd_quot_of_fract_nonzero: fixes x :: "'a fract" shows "snd (quot_of_fract x) \<noteq> (0::'a)"
decide_exis_lem_helper: fixes fml_struct :: "nat fml"   and polys :: "rat poly list"   and fml :: "rat poly fml" assumes "(fml_struct, polys) = convert fml" shows "(\<exists>x. lookup_sem fml_struct (map (\<lambda>p. rpoly p x) polys)) = decide_existential fml"
inner_loop2_def: fixes dst :: "'c"   and succl :: "'c list"   and u :: "'d"   and PRED :: "'c \<Rightarrow> 'd option"   and N :: "'c set" shows "inner_loop2 dst succl u PRED N \<equiv> nfoldli succl (\<lambda>(f, uu_, uu_). \<not> f) (\<lambda>v (f, PRED, N). if PRED v \<noteq> None then return (f, PRED, N) else let PRED = PRED(v \<mapsto> u) in assert (v \<notin> N) \<bind> (\<lambda>_. let N = insert v N in return (v = dst, PRED, N))) (False, PRED, N)"
dom_inv: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.dom (\<phi>.B.inv f) = \<phi>.B.cod f"
asIMOD_igSwapIGOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapIGOp (asIMOD SEM)"
IH_5: fixes a :: "'c"   and b :: "'d" assumes "a \<in> insert a' A"   and "b \<in> B" shows "g a b \<in> carrier G"
sorted_ins_list: fixes xs :: "'b list"   and x :: "'b" assumes "sorted xs" shows "sorted (ins_list x xs)"
UNext_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "UNext ch mm rs p \<equiv> RNext ch mm rs p \<or> MemFail ch rs p"
a_consistent_if\<^sub>2: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and a' :: "Arity"   and as' :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, a' # as') (\<Gamma>, Bool b, Alts e1 e2 # S)" shows "ArityAnalysisLetSafe.a_consistent Aexp (ae, a', as') (\<Gamma>, if b then e1 else e2, S)"
d: fixes x :: "complex"   and x' :: "complex" assumes "x \<in> cball 0 (B + 1)"   and "x' \<in> cball 0 (B + 1)"   and "dist x' x < d" shows "cmod (exp x' - exp x) < e"
signof_def: fixes p :: "nat \<Rightarrow> nat" shows "signof p = (if sign p = 1 then 1::'a else - (1::'a))"
abs_eq: fixes y :: "16 word" shows "Abs_uint16' y = Abs_uint16 y"
REFC: fixes it' :: "'Sa set"   and it :: "'S set"   and \<sigma>' :: "'b"   and \<sigma> :: "'a" assumes "it' = \<alpha> ` it"   and "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "\<Phi>'' it \<sigma> it' \<sigma>'"   and "\<Phi> it \<sigma>"   and "(\<sigma>, \<sigma>') \<in> R" shows "c \<sigma> = c' \<sigma>'"
neg_cong: fixes x :: "int" shows "\<ominus> (x mod m) = - x mod m"
pair_sym_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_sym_digraph G \<equiv> pair_wf_digraph G \<and> pair_sym_digraph_axioms G"
sim: shows "\<forall>i<length ts\<^sub>s\<^sub>b. \<exists>\<O>' \<D>' \<R>'. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts ! i in ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, [], \<D>', \<O>', \<R>')"
l_get_child_nodes_axioms: shows "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
one_hop: fixes \<xi> :: "state"   and \<xi>' :: "state"   and ip :: "nat" assumes "\<forall>dip\<in>kD (rt \<xi>). the (nhop (rt \<xi>) dip) \<in> kD (rt \<xi>)"   and "\<xi>' = \<xi>\<lparr>rt := update (rt \<xi>) ip (0, unk, val, Suc 0, ip)\<rparr>" shows "\<forall>dip\<in>kD (rt \<xi>). the (nhop (update (rt \<xi>) ip (0, unk, val, Suc 0, ip)) dip) = ip \<or> the (nhop (update (rt \<xi>) ip (0, unk, val, Suc 0, ip)) dip) \<in> kD (rt \<xi>)"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>* (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = f__" and "a3 = ms__" and "a4 = s__" and "a5 = []" and "a6 = ms__" and "a7 = s__" and "length ms__ = length s__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as' :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = last as # as'" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" and "S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s'')"
rInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and C :: "'d"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (M\<lparr>\<lambda>*xvec N\<rparr>.P) (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])"
simps_8: fixes x1 :: "'a"   and x2 :: "'a stream" shows "i.R (x1 ## x2) = insert x1 (i.R x2)"
comp_null_1: fixes f :: "'b" shows "J.comp J.null f = J.null"
unique_orbit: fixes x :: "'c"   and y :: "'c" assumes "x \<in> E"   and "y \<in> E" shows "\<exists>g\<in>carrier G. \<phi> g x = y"
bounds: fixes i :: "nat" shows "AE x in M. u i x \<le> w x"
n_p: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration" assumes "P c0"   and "next_propagate c0 c1" shows "P c1"
assms: shows "v \<in> statesDecls (states prog !! pState.idx p)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> bankerb_gets"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> bankerb_gets; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> bankerb_gets; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 A B. \<lbrakk>evs1 \<in> bankerb_gets; P evs1\<rbrakk> \<Longrightarrow> P (Says A Server \<lbrace>Agent A, Agent B\<rbrace> # evs1)"   and "\<And>evs2 K A B. \<lbrakk>evs2 \<in> bankerb_gets; P evs2; Key K \<notin> used evs2; K \<in> symKeys; Gets Server \<lbrace>Agent A, Agent B\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says Server A (Crypt (shrK A) \<lbrace>Number (CT evs2), Agent B, Key K, Crypt (shrK B) \<lbrace>Number (CT evs2), Agent A, Key K\<rbrace>\<rbrace>) # evs2)"   and "\<And>evs3 A Tk B K Ticket. \<lbrakk>evs3 \<in> bankerb_gets; P evs3; Gets A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs3; Says A Server \<lbrace>Agent A, Agent B\<rbrace> \<in> set evs3; \<not> expiredK Tk evs3\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Ticket, Crypt K \<lbrace>Agent A, Number (CT evs3)\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 B Tk A K Ta. \<lbrakk>evs4 \<in> bankerb_gets; P evs4; Gets B \<lbrace>Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>, Crypt K \<lbrace>Agent A, Number Ta\<rbrace>\<rbrace> \<in> set evs4; \<not> expiredK Tk evs4; \<not> expiredA Ta evs4\<rbrakk> \<Longrightarrow> P (Says B A (Crypt K (Number Ta)) # evs4)"   and "\<And>evso A Tk B K Ticket. \<lbrakk>evso \<in> bankerb_gets; P evso; Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evso; expiredK Tk evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Number Tk, Key K\<rbrace> # evso)" shows "P x"
wf_sig_iff_exe_wf_sig': fixes \<Sigma> :: "exesignature" assumes "exe_sig_conds \<Sigma>" shows "wf_sig (translate_signature \<Sigma>) = exe_wf_sig \<Sigma>"
prod_gcd_lcm_nat: fixes m :: "nat"   and n :: "nat" shows "m * n = gcd m n * lcm m n"
g: shows "g = (\<lambda>i. snd (f (j + i)))"
degree_prod: shows "degree (prod m {i. i < n}) = degree u"
comp_closed: fixes f :: "'a"   and g :: "'a" assumes "Arr f"   and "Arr g"   and "C.cod f = C.dom g" shows "Arr (C g f)"
natToVertexList_removeNones: fixes f :: "face"   and v :: "nat"   and es :: "nat list" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "incrIndexList es |es| |vertices f|" shows "filter (\<lambda>x. x \<in> set (removeNones (natToVertexList v f es))) (verticesFrom f v) = removeNones (natToVertexList v f es)"
abs_reach_def: fixes \<Phi> :: "'b ltl_prop_equiv_quotient" shows "af_G_abs.abs_reach \<Phi> = {foldl af_G_abs.f_abs \<Phi> w |w. True}"
col_rank_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "col_rank A = vec.dim (col_space A)"
simps_9: fixes x31 :: "'a itrm"   and x32 :: "'a itrm"   and x2 :: "dB" shows "x31 \<diamondop> x32 \<noteq> Pure x2"
orderE: fixes b :: "'a \<times> 'b"   and a :: "'a \<times> 'b" assumes "less_eq_prod b a" obtains "a = sup_prod a b"
real_scaleR_def: fixes a :: "real"   and x :: "real" shows "a *\<^sub>R x = a * x"
nonrec_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and y :: "Vars.var" assumes "atom ` domA \<Gamma> \<sharp>* x"   and "atom ` domA \<Gamma> \<sharp>* y" shows "nonrec \<Gamma>[x::h=y] = nonrec \<Gamma>"
bdia_export2: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "r p \<cdot> \<langle>x| q = \<langle>x \<cdot> r p| q"
norm_exp_i_times: fixes y :: "real" shows "cmod (iexp y) = 1"
nat_mirror_le: fixes a :: "nat"   and b :: "nat"   and l :: "nat"   and r :: "nat" assumes "a \<le> b" shows "nat_mirror b l r \<le> nat_mirror a l r"
vector_component_4: fixes c :: "'a"   and y :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(c *s y) $h i = c * y $h i"
winding_number_loops_linear_eq: fixes g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path g"   and "path h"   and "pathfinish g = pathstart g"   and "pathfinish h = pathstart h"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)" shows "winding_number h z = winding_number g z"
hom_0_iff: fixes x :: "'b mod_ring" shows "(to_int_mod_ring x = 0) = (x = 0)"
c''_aux: fixes i :: "nat" assumes "i \<le> N" shows "c' i = fps_to_fls (c''_aux i)"
blocking_on_def: fixes ds :: "D2 set"   and X :: "X4 set"   and h :: "H1"   and X' :: "X4 set" shows "StableNoDecomp.blocking_on ds X h X' = (X' \<noteq> CX4h h X \<and> X' = CX4h h (X \<union> X') \<and> X' \<subseteq> StableNoDecomp.CD_on ds (X \<union> X'))"