hcju/isabelleps huggingface dataset:
nhop_of_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flag, hops, nhip, {})" shows "the (nhop (update rt dip (dsn, dsk, flag, hops, nhip, {})) dip) = nhip"
minus_sq_mtx_def: shows "(-) \<equiv> map_fun to_vec (map_fun to_vec to_mtx) (-)"
restrict_map_disj: fixes S :: "'c set"   and T :: "'c set"   and h :: "'c \<Rightarrow> 'd option" assumes "S \<inter> T = {}" shows "h |` S \<bottom> h |` T"
pgwt_wellformed: fixes t :: "('fun, 'var) Term.term" assumes "public_ground_wf_term t" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
sup_compl_top_left2: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (sup (uminus x) y) = top"
hyps: fixes gpv :: "('a, 'call, 'ret) gpv" shows "expectation_gpv' gpv \<le> expectation_gpv 1 \<I> f gpv"
simps_6: fixes x :: "name"   and P :: "('d, 'e, 'f) psi" shows "guarded (\<lparr>\<nu>x\<rparr>P) = guarded P"
root_rat_reform: fixes x :: "rat"   and a :: "int"   and b :: "int"   and p :: "nat" assumes "quotient_of x = (a, b)" shows "root p (real_of_rat x) = root p (real_of_int (a * b ^ (p - 1))) / real_of_int b"
tensor_mat_eval: fixes i :: "nat"   and j :: "nat"   and m1 :: "'b mat"   and m2 :: "'b mat" assumes "i < d"   and "j < d" shows "tensor_mat m1 m2 $$ (i, j) = m1 $$ (encode1 i, encode1 j) * m2 $$ (encode2 i, encode2 j)"
A: shows "{} \<turnstile> \<alpha> IMP \<beta>"
null_space_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "null_space A = {x. A *v x = 0}"
one_strongly_connected_iff_3_eq: fixes x :: "'a" shows "one_strongly_connected x = (x ; 1 ; x = x\<^sup>T\<^sup>+)"
sorted_list_of_set_inj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "inj_on (linorder.sorted_list_of_set less_eq) (Collect finite)"
uses_in_vars: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" assumes "v \<in> uses' g n" shows "v \<in> braun_ssa.vars g"
SUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "\<Squnion> (f ` A) \<le> u"
F_eq: fixes n :: "nat" shows "F n = 2 * (\<Sum>k = 1..n. real (k - 1) / real (k * (k + 1)))"
step1_mat_size: shows "reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)) \<in> carrier_mat (length signs) (length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))))"
io_reduction_on_subset: fixes M1 :: "('d, 'e, 'f) FSM"   and T :: "'d list set"   and M2 :: "('d, 'e, 'g) FSM"   and T' :: "'d list set" assumes "M1 \<preceq>\<lbrakk>T\<rbrakk> M2"   and "T' \<subseteq> T" shows "M1 \<preceq>\<lbrakk>T'\<rbrakk> M2"
Lambert_W'_ge_zero: fixes x :: "real" assumes "0 \<le> x" shows "Lambert_W' x = - 1"
stable_pair_on_stable_no_blocking_on: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "stable_no_blocking_on ds (match XD_XH)"
has_integral': fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and S :: "'c set" shows "(f has_integral i) S = (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'c) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> S then f x else (0::'d)) has_integral z) (cbox a b) \<and> norm (z - i) < e))"
hyp: shows "a + b = b \<and> s \<oplus> t = t"
extend_set_UNIV_eq: shows "extend_set h UNIV = UNIV"
subset_imp_lepoll: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<lesssim> B"
T_composition_borel_1: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f (T x)) \<in> borel_measurable M"
absorb2: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" shows "inf a b = b"
convex_rel_interior_if: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "z \<in> rel_interior S" shows "\<forall>x\<in>affine hull S. \<exists>m>1. \<forall>e. 1 < e \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
v_def: shows "v = birkhoff_sum (\<lambda>x. - subcocycle_lim u x)"
cases: fixes x :: "('c, 'd) t" obtains xs :: "('c \<times> 'd) list" where "x = T xs"
1: shows "bprv (all yy (imp (cnj \<chi> (NN \<langle>\<phi>\<rangle> (Var yy))) (subst \<chi> \<langle>neg \<phi>\<rangle> yy)))"
sound_modes: shows "\<forall>mem. INIT mem \<longrightarrow> sound_mode_use (cmds, mem)"
cases: fixes a1 :: "('fenv_std, 'a, 'b) state"   and a2 :: "('fenv_inca, 'a, 'b) state" assumes "a1 \<sim> a2" obtains F1 :: "'fenv_std" and F2 :: "'fenv_inca" and H :: "'a" and st :: "'b list" where "a1 = State F1 H st" and "a2 = State F2 H st" and "rel_fundefs (Fstd_get F1) (Finca_get F2)"
get_owner_document_wf_get_root_node_wf_is_l_get_owner_document_wf_get_root_node_wf: shows "l_get_owner_document_wf_get_root_node_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_root_node Shadow_DOM.get_owner_document"
\<tau>inv_axioms: shows "\<tau>inv trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"
edka_imp_refine: fixes am :: "nat \<Rightarrow> nat list" shows "hn_refine emp (edka_imp c s t N am) emp (is_rflow N) (edka5 am)"
additive_increasing: fixes f :: "('i \<Rightarrow> 'a) set \<Rightarrow> ennreal" assumes "positive generator f"   and "Measure_Space.additive generator f" shows "increasing generator f"
finprod_multf: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> carrier G"   and "g \<in> A \<rightarrow> carrier G" shows "(\<Otimes>x\<in>A. f x \<otimes> g x) = finprod G f A \<otimes> finprod G g A"
finite_wqo_on: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite A"   and "reflp_on P A"   and "transp_on P A" shows "wqo_on P A"
semiring_1_cancel_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semiring_1_cancel plus minus zero one times \<equiv> (class.cancel_comm_monoid_add plus minus zero \<and> class.monoid_mult one times) \<and> class.semiring plus times \<and> class.zero_neq_one one zero"
inter_aform_plane_def: fixes prec :: "nat"   and Xs :: "'b \<times> 'b pdevs"   and sctn :: "'b sctn" shows "inter_aform_plane prec Xs sctn = inter_aform_plane_ortho_nres prec Xs (normal sctn) (pstn sctn) \<bind> (\<lambda>cxs. case cxs of None \<Rightarrow> SUCCEED | Some cxs \<Rightarrow> if normal sctn \<in> set Basis_list then ASSERT (index Basis_list (normal sctn) < DIM('b)) \<bind> (\<lambda>_. RETURN (project_ncoord_aform cxs (index Basis_list (normal sctn)) (pstn sctn))) else if - normal sctn \<in> set Basis_list then ASSERT (index Basis_list (- normal sctn) < DIM('b)) \<bind> (\<lambda>_. RETURN (project_ncoord_aform cxs (index Basis_list (- normal sctn)) (- pstn sctn))) else SUCCEED)"
C\<^sub>1': shows "\<^bold>\<forall>x xa xb. E (x \<cdot> xa) \<^bold>\<leftarrow> E (xa \<cdot> xb) \<^bold>\<and> E (x \<cdot> (xa \<cdot> xb))"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'a \<Rightarrow> 'd" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
Pre_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" shows "Pre D a = (SOME x. x \<in> carrier D \<and> x \<prec>\<^bsub>D\<^esub> a \<and> \<not> (\<exists>y\<in>carrier D. x \<prec>\<^bsub>D\<^esub> y \<and> y \<prec>\<^bsub>D\<^esub> a))"
alt_env_def: fixes alt :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m1 :: "('d, 'e) envT"   and m2 :: "('d, 'e) envT" shows "alt_env alt m1 m2 = EnvT (\<lambda>r. alt (run_env m1 r) (run_env m2 r))"
L3_ControlSystem: fixes s :: "nat \<Rightarrow> nat list"   and i :: "nat" assumes "ControlSystemArch s" shows "hd (s i) \<le> 800"
scale_cancel_left: fixes a :: "'c"   and x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "(a *s x = a *s y) = (x = y \<or> a = (0::'c))"
defs_2: shows "B = filter (\<lambda>i. \<not> P i) [0..<n]"
E: fixes k :: "nat" shows "\<forall>x\<in>{0..}. cmod (if x \<in> {0..real k} then complex_of_real x powr (z - 1) * (1 - complex_of_real x / of_nat k) ^ k else 0) \<le> h x"
single_iface: fixes I :: "qtname" shows "is_iface tprg I = (I = HasFoo)"
vars1'_def: fixes vars1 :: "nat set"   and vars2 :: "nat set" assumes "partial_state2 vars1 vars2" shows "partial_state2.vars1' vars1 vars2 = ind_in_set (partial_state2.vars0 vars1 vars2) ` vars1"
asym: fixes a :: "'c word"   and b :: "'c word" assumes "a <s b"   and "b <s a" shows "False"
mL_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and I :: "'c set"   and j :: "nat" shows "mL K P I j = tna (LI K (\<nu>\<^bsub>K P j\<^esub>) I)"
mat_append_row_row: fixes A :: "'a mat"   and c :: "'a vec" assumes "dim_col A = dim_vec c" shows "row (mat_append_row A c) (dim_row A) = c"
prems: shows "\<forall>a\<in>insert x (set (xs @ [y])). case a of (i, p) \<Rightarrow> \<forall>a\<in>insert x (set (xs @ [y])). case a of (i', p') \<Rightarrow> p = p' \<longrightarrow> i = i'"
proper_map_on_empty: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "topspace X = {}" shows "proper_map X Y f"
group_subgroup_generated: fixes S :: "'a set" shows "Group.group (subgroup_generated G S)"
rel_inject_5: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x5 :: "'a \<times> nat"   and y5 :: "'b \<times> nat" shows "rel_SDG_node R (Actual_out x5) (Actual_out y5) = rel_prod R (=) x5 y5"
vimage_Compl: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set" shows "f -` (- A) = - f -` A"
that: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "\<Union> (range A) = space M"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>"   and "incseq A" shows "thesis"
fs_cont: fixes k :: "nat" assumes "k \<le> N" shows "continuous_on {real a..} (fs k)"
sorted_insort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.insort_key less_eq (\<lambda>x. x) x xs) = linorder.sorted less_eq xs"
pivot_and_update_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'b"   and s :: "('a, 'b) state" assumes "PivotUpdate eq_idx_for_lvar pivot update" shows "PivotUpdate.pivot_and_update pivot update x\<^sub>i x\<^sub>j c s \<equiv> update x\<^sub>i c (pivot x\<^sub>i x\<^sub>j s)"
shift2_invD_3: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "shift2_inv qs" shows "exact_decomp (Suc m) qs"
le_max_list: fixes xs :: "'b list"   and a :: "'b"   and b :: "'b" assumes "\<And>a b. \<lbrakk>a \<in> set xs; b \<in> set xs\<rbrakk> \<Longrightarrow> a \<le> b \<or> b \<le> a"   and "a \<le> b"   and "b \<in> set xs" shows "a \<le> Missing_Lemmas.max_list xs"
counit_simps_4: shows "src \<epsilon>\<^sub>0 = trg e\<^sub>0"
assms_3: shows "u \<turnstile>\<^bsub>v,n\<^esub> M"
simps: fixes a :: "'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" shows "(a \<in> smallest_ccdi_sets \<Omega> M) = ((\<exists>aa. a = aa \<and> aa \<in> M) \<or> (\<exists>aa. a = \<Omega> - aa \<and> aa \<in> smallest_ccdi_sets \<Omega> M) \<or> (\<exists>A. a = \<Union> (range A) \<and> range A \<in> Pow (smallest_ccdi_sets \<Omega> M) \<and> A 0 = {} \<and> (\<forall>x. A x \<subseteq> A (Suc x))) \<or> (\<exists>A. a = \<Union> (range A) \<and> range A \<in> Pow (smallest_ccdi_sets \<Omega> M) \<and> disjoint_family A))"
elims: fixes x :: "arctic"   and y :: "int" assumes "get_arctic_num x = y" obtains n :: "int" where "x = Num_arc n" and "y = n"   | "x = MinInfty" and "y = undefined"
cart_product_single_intersect: fixes x1 :: "'a"   and x2 :: "'a"   and A :: "'b multiset"   and B :: "'b multiset" assumes "x1 \<noteq> x2" shows "{#x1#} \<times># A \<inter># {#x2#} \<times># B = {#}"
observe_execution_step_def: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and as :: "(String.literal \<times> value list) list" shows "observe_execution e s r ((l, i) # as) = (case step e s r l i of None \<Rightarrow> [] | Some (t, s', p, r') \<Rightarrow> p # observe_execution e s' r' as)"
igSubstIGOpSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGOpSTR MOD \<equiv> (\<forall>ys y y1 delta inp binp. igSubst MOD ys (igVar MOD ys y1) y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys (igVar MOD ys y1) y inp) (igSubstBinp MOD ys (igVar MOD ys y1) y binp)) \<and> (\<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp))"
COD_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "monoidal_language.Cod C (free_monoidal_category.rep C (free_monoidal_category.mkarr C t)) = monoidal_language.Cod C t"
undual_minus_eq: fixes x :: "'a dual"   and y :: "'a dual" shows "undual (x - y) = - (undual y - undual x)"
multiplicity_normalize_right: fixes p :: "'a"   and x :: "'a" shows "multiplicity p (normalize x) = multiplicity p x"
FOREACH: shows "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')"
add_pCons: fixes a :: "'c"   and p :: "'c poly"   and b :: "'c"   and q :: "'c poly" shows "pCons a p + pCons b q = pCons (a + b) (p + q)"
lossless_weight_spmfD: fixes p :: "'d spmf" assumes "lossless_spmf p" shows "weight_spmf p = 1"
cempty_iff: fixes c :: "'d" shows "cin c cempty = False"
compactE: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and \<T> :: "'a set set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "S \<subseteq> \<Union> \<T>"   and "\<And>B. B \<in> \<T> \<Longrightarrow> open B" obtains \<T>' :: "'a set set" where "\<T>' \<subseteq> \<T>" and "finite \<T>'" and "S \<subseteq> \<Union> \<T>'"
surjective: fixes r :: "('c, 'd) boolean_algebra_scheme" shows "r = \<lparr>carrier = carrier r, eq = (.=\<^bsub>r\<^esub>), le = (\<sqsubseteq>\<^bsub>r\<^esub>), compl = \<^bold>-\<^bsub>r\<^esub>, \<dots> = boolean_algebra.more r\<rparr>"
is_regular_has_right_codom: 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 U :: "'c set set"   and \<pp> :: "'c set"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit"   and "U \<subseteq> comm_ring.spectrum R addition multiplication zero unit"   and "\<pp> \<in> U"   and "comm_ring.is_regular R addition multiplication zero unit s U" shows "s \<pp> \<in> R\<setminus>\<pp> \<^sup>\<inverse> R \<^bsub>addition multiplication zero\<^esub>"
dg_prod_ObjD_2: fixes a :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>" shows "\<D>\<^sub>\<circ> a = I"
llexord_LNil_right: fixes ys :: "'c llist"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist" assumes "lnull ys" shows "llexord r xs ys = lnull xs"
validn_valid: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "\<forall>n. G \<Turnstile>\<^sub>n c : A" shows "G \<Turnstile> c : A "
HEndPhase0_blocksOf2: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv2c_inner s p" shows "allBlocksRead s p \<subseteq> blocksOf s p"
tau_star_nu: fixes x :: "'a" shows "\<tau> (\<nu> x\<^sup>\<star>) = (0::'a)"
acc_point_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" shows "acc_point s k = (if k = 0 then 0 else res_seq s k 0 k)"
2: shows "\<forall>i j. i < row_length A1 * row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> (A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = scalar_product (row A1 (i div row_length B1)) (col A2 (j div length B2)) * scalar_product (row B1 (i mod row_length B1)) (col B2 (j mod length B2))"
le_csucc: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> \<le> csucc \<kappa>"
subseqs_subformulas\<^sub>\<nu>_list: fixes Y :: "'a ltln set"   and \<phi> :: "'a ltln" shows "(Y \<subseteq> subformulas\<^sub>\<nu> \<phi>) = (\<exists>ys. Y = set ys \<and> ys \<in> set (subseqs (subformulas\<^sub>\<nu>_list \<phi>)))"
usubstr_cons: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and A :: "fml"   and B :: "fml list"   and C :: "fml" assumes "usubstr \<sigma> (A # B, C) \<noteq> None" shows "the (usubstr \<sigma> (A # B, C)) = (the (usubstappf \<sigma> Denotational_Semantics.allvars A) # fst (the (usubstr \<sigma> (B, C))), snd (the (usubstr \<sigma> ([A], C))))"
P: fixes l :: "'a" assumes "is_glb L l A" shows "P l"
finsert_commute: fixes x :: "'a"   and y :: "'a"   and A :: "'a fset" shows "finsert x (finsert y A) = finsert y (finsert x A)"
th1: shows "bound0 (Parametric_Ferrante_Rackoff.disj (minusinf (simpfm p)) (Parametric_Ferrante_Rackoff.disj (plusinf (simpfm p)) (evaldjf (simpfm \<circ> msubst (simpfm p)) (alluopairs (remdups (uset (simpfm p)))))))"
second_vertices_new_path: fixes xs :: "'a list" assumes "hd (tl xs) \<notin> Q.second_vertices" shows "xs \<notin> Q"
x_notin: shows "x \<notin> set (vwalk_to_vpath xs)"
a_eq: shows "a = rev [0..<IArray.length (vec_to_iarray A)] ! ia"
assms_1: shows "p = {l}unicast(fip, fmsg) . q1 \<triangleright> q2"
owns_def: fixes G :: "igraph"   and l :: "location"   and a :: "actor"   and d :: "(actor \<times> actor set) \<times> nat" shows "owns G l a d \<equiv> owner d = a"
rel_option_None2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a option" shows "rel_option P x None = (x = None)"
Suc_Suc_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>n. \<lbrakk>P n; P (Suc n)\<rbrakk> \<Longrightarrow> P (Suc (Suc n))" shows "P n"
mult_commute: fixes a :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
preferred_alts_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "preferred_alts R x = {y. R x y}"
rel_pmf_iff_measure: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "symp R"   and "transp R" shows "rel_pmf R p q = (rel_set R (set_pmf p) (set_pmf q) \<and> (\<forall>x\<in>set_pmf p. \<forall>y\<in>set_pmf q. R x y \<longrightarrow> measure_pmf.prob p {x. R x y} = measure_pmf.prob q {y. R x y}))"
fut_spec_def: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and LESS :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and assigns :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'd"   and nvars :: "'c \<Rightarrow> 'b"   and Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and CONS :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and SNOC :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and Length :: "'c \<Rightarrow> nat"   and extend :: "'a \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'e"   and size :: "'e \<Rightarrow> 'b"   and zero :: "'b \<Rightarrow> 'e"   and alphabet :: "'b \<Rightarrow> 'e list"   and eval :: "'d \<Rightarrow> nat \<Rightarrow> bool"   and downshift :: "'d \<Rightarrow> 'd"   and upshift :: "'d \<Rightarrow> 'd"   and add :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and cut :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and len :: "'d \<Rightarrow> nat"   and restrict :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and Restrict :: "'a \<Rightarrow> nat \<Rightarrow> ('f, 'a) aformula"   and lformula0 :: "'f \<Rightarrow> bool"   and FV0 :: "'a \<Rightarrow> 'f \<Rightarrow> nat set"   and find0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> bool"   and wf0 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and decr0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> 'f"   and satisfies0 :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and nullable0 :: "'f \<Rightarrow> bool"   and lderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and rderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and b :: "bool"   and idx :: "'b"   and \<psi> :: "('f, 'a) aformula"   and \<phi>\<Phi> :: "'g \<times> ('f, 'a) aformula list" assumes "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 satisfies0 nullable0 lderiv0 rderiv0" shows "Formula.fut_spec SUC nvars Extend SNOC Length zero len wf0 satisfies0 b idx \<psi> \<phi>\<Phi> \<equiv> (\<forall>\<phi>\<in>set (snd \<phi>\<Phi>). Formula_Operations.wf SUC wf0 idx \<phi>) \<and> (\<forall>\<AA>. nvars \<AA> = idx \<longrightarrow> (if b then (\<exists>k. Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ P. (\<le>) (len P)) ((SNOC (zero idx) ^^ k) \<AA>) \<psi>) = (\<exists>\<phi>\<in>set (snd \<phi>\<Phi>). Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ P. (\<le>) (len P)) \<AA> \<phi>) else (\<forall>k. Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ P. (\<le>) (len P)) ((SNOC (zero idx) ^^ k) \<AA>) \<psi>) = (\<forall>\<phi>\<in>set (snd \<phi>\<Phi>). Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ P. (\<le>) (len P)) \<AA> \<phi>)))"
IH: fixes u :: "nat" shows "pathVertices u (es @ p2) = butlast (pathVertices u es) @ pathVertices (last (pathVertices u es)) p2"
assms_5: fixes b :: "bool" assumes "y = Bool b" shows "P"
span_eq: fixes S :: "'a poly set"   and T :: "'a poly set" shows "(poly_vs.span S = poly_vs.span T) = (S \<subseteq> poly_vs.span T \<and> T \<subseteq> poly_vs.span S)"
hyps: assumes "\<forall>y. y \<in> set xs \<longrightarrow> (0::'a) < y" shows "(0::'a) < prod_list xs"
is_ub_thelub: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and x :: "nat" assumes "class.cpo below"   and "po.chain below S" shows "below (S x) (po.lub below (range S))"
l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf ShadowRootClass.known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs"