hcju/isabelleps huggingface dataset:
integrable_affinity: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and m :: "real"   and c :: "'a" assumes "f integrable_on cbox a b"   and "m \<noteq> 0" shows "(\<lambda>x. f (m *\<^sub>R x + c)) integrable_on (\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b"
add_le_imp_le_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus a c) (plus b c)" shows "less_eq a b"
valid_conj: fixes ds :: "'b set"   and R :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "valid ds (\<lambda>ds fp. R ds fp \<and> P ds fp \<and> Q ds fp) P"   and "valid ds (\<lambda>ds fp. R ds fp \<and> P ds fp \<and> Q ds fp) Q" shows "valid ds R (\<lambda>ds fp. P ds fp \<and> Q ds fp)"
merge_status_eq_iff_2: fixes a :: "status" shows "(merge_status a SUCCESS = FOUND) = (a = FOUND)"
const_of_i32: fixes cs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "const_list cs"   and "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_i32]" shows "\<exists>c. cs = [$C ConstInt32 c]"
distinct_5: fixes x41 :: "expr"   and x42 :: "cmd"   and x43 :: "cmd" shows "Skip \<noteq> if (x41) x42 else x43"
lm104: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>X. f x = g x" shows "sum f X = sum g X"
1: shows "set hs \<times> (set gs \<union> set bs) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
S: shows "\<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in net. f x \<in> S)"
evaln_type_sound: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and n :: "nat"   and v :: "vals"   and s1 :: "state"   and accC :: "qtname"   and L :: "lenv"   and T :: "tys"   and A :: "assigned" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "s0\<Colon>\<preceq>(G, L)"   and "wf_prog G" shows "s1\<Colon>\<preceq>(G, L) \<and> (normal s1 \<longrightarrow> G,L,snd s1\<turnstile>t\<succ>v\<Colon>\<preceq>T) \<and> error_free s0 = error_free s1"
apath_simps_1: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and v :: "'b" assumes "wf_digraph G" shows "pre_digraph.apath G u [] v = (u = v \<and> u \<in> verts G)"
oldDefs_def: fixes g :: "'g"   and n :: "'node" shows "oldDefs g n = var g ` defs g n"
countable_INT'': fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c set" assumes "UNIV \<in> sigma_sets UNIV (range lessThan)"   and "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sigma_sets UNIV (range lessThan)" shows "\<Inter> (F ` I) \<in> sigma_sets UNIV (range lessThan)"
light_year: shows "light_year \<cong>\<^sub>Q \<^bold>c \<^bold>\<cdot> julian_year"
xfer_has_ticket: fixes ts :: "'a \<Rightarrow> thread"   and t :: "'a"   and tki :: "nat" assumes "has_ticket (map_ticket (\<lambda>t. t mod N) (ts t)) tki" obtains tk :: "nat" where "tki = tk mod N" and "has_ticket (ts t) tk"
term_variants_Fun: fixes T :: "('c, 'd) Term.term list"   and S :: "('c, 'd) Term.term list"   and P :: "'c \<Rightarrow> 'c list"   and f :: "'c" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)" shows "term_variants_pred P (Fun f T) (Fun f S)"
rightreg_scalar_mult_def: fixes r :: "'r"   and f :: "'r \<Rightarrow> 'a" shows "r \<currency> f = (\<lambda>x. if x \<in> R then f (x * r) else (0::'a))"
GF_advice_minus_size: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and X :: "'b ltln set" assumes "size \<phi> \<le> size \<psi>"   and "\<phi> \<noteq> \<psi>" shows "\<phi>[X - {\<psi>}]\<^sub>\<nu> = \<phi>[X]\<^sub>\<nu>"
Let_def: fixes s :: "'l"   and f :: "'l \<Rightarrow> 'k" shows "Let s f \<equiv> f s"
H_2: shows "z \<in> extended_distance_set \<and> t \<in> extended_distance_set \<and> extended_distance z t = 0"
simps: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'a literal multiset" shows "ord_resolve a1 a2 a3 a4 a5 = (\<exists>CAs n Cs AAs As D. a1 = CAs \<and> a2 = D + negs (mset As) \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<Sum>\<^sub># (mset Cs) + D \<and> length CAs = n \<and> length Cs = n \<and> length AAs = n \<and> length As = n \<and> n \<noteq> 0 \<and> (\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)) \<and> (\<forall>i<n. AAs ! i \<noteq> {#}) \<and> (\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i) \<and> eligible As (D + negs (mset As)) \<and> (\<forall>i<n. strictly_maximal_wrt (As ! i) (Cs ! i)) \<and> (\<forall>i<n. S (CAs ! i) = {#}))"
CFGExit_wf_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g" shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit \<equiv> CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val \<and> CFGExit sourcenode targetnode kind valid_edge Entry Exit \<and> CFGExit_wf_axioms Def Use Exit"
SBHFence: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Fence # is, \<theta>, [], m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, [], m, False, \<O>, Map.empty, \<S>)"
left_adjoint_functor_axioms: shows "left_adjoint_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
normalize_ports_generic_preserves_normalized_not_has_disc: fixes m :: "'b common_primitive match_expr"   and disc2 :: "'b common_primitive \<Rightarrow> bool"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc disc2 m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc (Prot a)"   and "\<forall>a. \<not> disc2 (C a)"   and "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"   and "m' \<in> set (normalize_ports_generic (normalize_positive_ports_step (disc, sel) C) (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one) m)" shows "\<not> has_disc disc2 m'"
pt2: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
sol: shows "(A *\<^sub>v x) $ i \<le> b $ i"
n_mult_lb1': fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<le> !x"
IH: fixes i :: "'b" assumes "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "dsafe (Tsubst (args i) \<sigma>)"
inf_import_p_eq: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - (x \<sqinter> y) = x \<sqinter> - y"
wffp_purgeIdle: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "purgeIdle sl \<noteq> []" shows "wffp (purgeIdle sl)"
infinite_card_of_diff_singl: fixes A :: "'a set"   and a :: "'a" assumes "infinite A" shows "|A - {a}| =o |A|"
nres_monad_laws_1: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b nres" shows "RETURN x \<bind> f = f x"
card_lessThan_enat: fixes k :: "nat" shows "card {..<enat k} = card {..<k}"
Seq_Skip: fixes A :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "A ;; Skip = A"
imp_nfoldli_simps_1: fixes c :: "'c \<Rightarrow> bool Heap"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and s :: "'c" shows "imp_nfoldli [] c f s = return s"
in_gen_check: fixes xs :: "nat list"   and s :: "nat"   and as :: "nat list" assumes "(xs, s) \<in> set (c1.gen_check as)" shows "length xs = length as \<and> s = as \<bullet> xs"
conj_lin: fixes p :: "fm"   and q :: "fm" assumes "isrlfm p"   and "isrlfm q" shows "isrlfm (Ferrack.conj p q)"
hom: shows "double_dual_iso \<in> hom G (Characters (Characters G))"
refine: fixes A :: "('c \<times> 'd) set"   and Xi :: "'c list"   and X :: "'d set"   and P_impl :: "'c \<Rightarrow> bool nres"   and P :: "'d \<Rightarrow> bool nres"   and I :: "'d \<Rightarrow> bool" assumes "single_valued A"   and "(Xi, X) \<in> \<langle>A\<rangle>list_wset_rel"   and "(P_impl, P) \<in> A \<rightarrow> \<langle>bool_rel\<rangle>nres_rel" shows "(WEAK_EX_list Xi P_impl, WEAK_EX\<^bsup>I\<^esup> X P) \<in> \<langle>bool_rel\<rangle>nres_rel"
f7: shows "rk {A', B', B, \<gamma>} = 2"
select_convs_5: fixes inpt :: "Proc \<Rightarrow> InputsOrNi"   and outpt :: "Proc \<Rightarrow> InputsOrNi"   and disk :: "Disk \<Rightarrow> Proc \<Rightarrow> DiskBlock"   and dblock :: "Proc \<Rightarrow> DiskBlock"   and phase :: "Proc \<Rightarrow> nat"   and disksWritten :: "Proc \<Rightarrow> Disk set"   and blocksRead :: "Proc \<Rightarrow> Disk \<Rightarrow> BlockProc set"   and allInput :: "InputsOrNi set"   and chosen :: "InputsOrNi"   and more :: "'a" shows "phase \<lparr>inpt = inpt, outpt = outpt, disk = disk, dblock = dblock, phase = phase, disksWritten = disksWritten, blocksRead = blocksRead, allInput = allInput, chosen = chosen, \<dots> = more\<rparr> = phase"
ev_shift: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and xl :: "'b list" assumes "ev \<phi> xs" shows "ev \<phi> (xl @- xs)"
map_iterator_rev_linord_I: fixes l0 :: "('a \<times> 'd) list"   and m :: "'a \<Rightarrow> 'd option"   and iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'd \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "distinct (map fst l0)"   and "m = map_of l0"   and "sorted (rev (map fst l0))"   and "iti = foldli l0" shows "map_iterator_rev_linord iti m"
Mp_ident_iff: fixes f :: "int poly" shows "(Mp f = f) = (\<forall>n. coeff f n \<in> {0..<p})"
simps: fixes A :: "'b mat"   and B :: "'b mat"   and i :: "nat"   and j :: "nat" shows "gauss_jordan_main A B i j = (let nr = dim_row A; nc = dim_col A in if i < nr \<and> j < nc then let aij = A $$ (i, j) in if aij = (0::'b) then case concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'b) then [i'] else []) [Suc i..<nr]) of [] \<Rightarrow> gauss_jordan_main A B i (Suc j) | i' # x \<Rightarrow> gauss_jordan_main (swaprows i i' A) (swaprows i i' B) i j else if aij = (1::'b) then let v = \<lambda>i. A $$ (i, j) in gauss_jordan_main (eliminate_entries v A i j) (eliminate_entries v B i j) (Suc i) (Suc j) else let iaij = inverse aij in gauss_jordan_main (multrow i iaij A) (multrow i iaij B) i j else (A, B))"
of_bool_def: fixes one :: "'b"   and zero :: "'b"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero p = (if p then one else zero)"
eqI_snth: fixes u :: "'b stream"   and v :: "'b stream" assumes "\<And>i. u !! i = v !! i" shows "u = v"
try_spmf_def: fixes p :: "'a spmf"   and q :: "'a spmf" shows "TRY p ELSE q = bind_pmf p (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
LC: shows "while pa_cond (pa_step H1 H2) (pa_initial H1 H2) = dwa.loop"
Ifun_star_of: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "star_of f \<star> star_of x = star_of (f x)"
all_edges_mono: fixes vs :: "nat set"   and ws :: "nat set" assumes "vs \<subseteq> ws" shows "all_edges vs \<subseteq> all_edges ws"
meet_lower1: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<sqsubseteq> x"
can_implies_arr: fixes f :: "'c local.term set" assumes "\<F>C.can f" shows "\<F>C.arr f"
Gr_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "BNF_Def.Gr A f = {(a, f a) |a. a \<in> A}"
h5: shows "\<forall>t1\<le>t. req t1 = []"
simps_35: fixes vd :: "char list"   and ve :: "aexp"   and vf :: "'b"   and v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b" shows "(vd ::= ve {vf} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
sat_plan_to_dimacs_works: fixes sv :: "sat_plan_variable" assumes "valid_state_var sv" shows "dimacs_to_var (var_to_dimacs sv) = sv"
simps: fixes \<sigma> :: "nat \<Rightarrow> bool"   and l1 :: "lit"   and l2 :: "lit" shows "sat_clause \<sigma> (Upair l1 l2) = (sat_lit \<sigma> l1 \<or> sat_lit \<sigma> l2)"
next_lins2_vertex: fixes v :: "node"   and \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list"   and S :: "node fset"   and G :: "diagram" assumes "Inl v # \<pi> \<in> lins2 S G"   and "v |\<notin>| S" shows "\<pi> \<in> lins2 ({|v|} |\<union>| S) G"
disj_matrices_add_le_zero: fixes A :: "'a matrix"   and B :: "'a matrix" assumes "disj_matrices A B" shows "(A + B \<le> 0) = (A \<le> 0 \<and> B \<le> 0)"
dim_span: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "vector_space scale" shows "vector_space.dim scale (module.span scale S) = vector_space.dim scale S"
SUP_upper2: fixes i :: "'c"   and A :: "'c set"   and u :: "'d"   and f :: "'c \<Rightarrow> 'd" assumes "i \<in> A"   and "u \<le> f i" shows "u \<le> \<Squnion> (f ` A)"
normalize_negative: fixes q :: "int"   and p :: "int" assumes "q < 0" shows "Rat.normalize (p, q) = Rat.normalize (- p, - q)"
vector_space_on_axioms: shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) Cod_g.map"
unfoldI: fixes P :: "pi"   and Q :: "pi" assumes "\<forall>s. P[
] \<leadsto><weakBisim> Q[
] \<and> Q[
] \<leadsto><weakBisim> P[
]" shows "P \<simeq>\<^sup>s Q"
run_eval_match: shows "\<exists>run_eval_match. \<forall>env v pes err_v s. evaluate_match True env s v pes err_v (run_eval_match env v pes err_v s)"
assms_5: shows "\<forall>\<^sub>F x in F. g x \<noteq> 0"
cont_Ndet1: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "((\<Squnion>i. Y i) |-| S) = (\<Squnion>i. Y i |-| S)"
bind_UNION: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "A \<bind> f = \<Union> (f ` A)"
m2: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "m y < m x"
fls_shifted_times_simps_2: fixes n :: "int"   and f :: "'a fls"   and g :: "'a fls" shows "fls_shift n f * g = fls_shift n (f * g)"
simps_1: shows "eval_ternary_And TernaryTrue TernaryTrue = TernaryTrue"
assms: shows "dist B (midpoint A C) = dist A C / 2"
has_vector_derivative_within_subset: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and S :: "real set"   and T :: "real set" assumes "(f has_vector_derivative f') (at x within S)"   and "T \<subseteq> S" shows "(f has_vector_derivative f') (at x within T)"
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ** y ** z = x ** (y ** z)"
mono_Un: fixes Retr :: "'b set \<Rightarrow> 'c set"   and theta :: "'b set"   and theta' :: "'b set" assumes "Bisim.mono Retr" shows "Retr theta \<union> Retr theta' \<subseteq> Retr (theta \<union> theta')"
not_iff_asym: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Y"   and "y \<in> Y" shows "(\<not> x \<sqsubseteq> y) = (y \<sqsubset> x)"
delete_conv: fixes k :: "'b"   and al :: "('b \<times> 'a) list"   and k' :: "'b" shows "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
fold_simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list"   and s :: "'c" shows "List.fold f (x # xs) s = List.fold f xs (f x s)"
ltl_prop_equiv_equivp: shows "equivp (\<sim>\<^sub>P)"
int_pow_mult: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and i :: "int"   and j :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> (i + j) = x [^]\<^bsub>G\<^esub> i \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> j"
dbm_add_not_inf: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<noteq> \<infinity>"   and "b \<noteq> \<infinity>" shows "dbm_add a b \<noteq> \<infinity>"
diamond_def: fixes r :: "comb \<Rightarrow> comb \<Rightarrow> bool" shows "diamond r \<equiv> \<forall>x y. r x y \<longrightarrow> (\<forall>y'. r x y' \<longrightarrow> (\<exists>z. r y z \<and> r y' z))"
order_iff: fixes b :: "'b"   and a :: "'b" shows "order.greater_eq a b = (a = sup a b)"
measure_notin_sets: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<notin> sets M" shows "Sigma_Algebra.measure M A = 0"
comm_semiring_isom_axioms: shows "comm_semiring_isom hom"
list_update_code_3: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat"   and y :: "'b" shows "(x # xs)[Suc i := y] = x # xs[i := y]"
DsI: fixes t :: "'a set"   and r :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> t \<Longrightarrow> Domainp A (r x)"   and "t \<subseteq> Collect (Domainp A)" shows "Domainp A (sum_with p z r t)"
IH_1: assumes "inv_max_hi l"   and "Sorted_Less.sorted (Tree2.inorder l)" shows "search l x = has_overlap (Tree2.set_tree l) x"
polyhedron_def: fixes A :: "'b mat"   and b :: "'b vec" shows "m.polyhedron A b = {x \<in> carrier_vec n. A *\<^sub>v x \<le> b}"
comp_id: fixes f :: "'a \<Rightarrow> 'b" shows "f \<circ> Fun.id = f"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_1: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, send\<langle>t\<rangle>) # A) = (l, receive\<langle>t\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
m_translate_def: shows "m_translate = (\<lambda>r\<in>carrier R. restrict ((\<otimes>) r) (carrier R))"
f_aggregate_1: fixes ag :: "'b list \<Rightarrow> 'b"   and xs :: "'b list" assumes "\<And>x. ag [x] = x" shows "f_aggregate xs (Suc 0) ag = xs"
closed_halfspace_le: fixes a :: "'b"   and b :: "real" shows "closed {x. a \<bullet> x \<le> b}"
delete_invalid_block_eq: fixes b :: "'a set" assumes "b \<notin># add_block bl" shows "fin.del_block b = add_block bl"
intro: fixes X :: "'b set set" assumes "\<forall>x\<in>X. finite x"   and "\<And>x y. \<lbrakk>x \<in> X; order.greater_eq x y\<rbrakk> \<Longrightarrow> y \<in> X" shows "SimplicialComplex X"
AndR_in_SNa: fixes M :: "trm"   and N :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname" assumes "SNa M"   and "SNa N" shows "SNa (AndR <a>.M <b>.N c)"
terminal_def: fixes b :: "'s" shows "S.terminal b \<equiv> S.ide b \<and> (\<forall>a. S.ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
fempty_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fset A {||} {||}"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>ip = ip r, sn = sn r, rt = rt r, rreqs = rreqs r, store = store r, msg = msg r, data = data r, dests = dests r, dip = dip r, oip = oip r, hops = hops r, dsn = dsn r, dsk = dsk r, osn = osn r, sip = sip r, handled = handled r, \<dots> = state.more r\<rparr>"
3: shows "bprv (all yy (imp (SS \<langle>\<chi>\<rangle> (Var yy)) (eql \<langle>\<psi>\<rangle> (Var yy))))"
normalize_Gcd: fixes A :: "'a set" shows "normalize (Gcd A) = Gcd A"
groupWIs3_default_policy_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs3_default_policy c rs = (let P = getParts rs; W = map getOneIp P; filterW = filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs; f = \<lambda>wi. let mtch_dsts = matching_dsts (getOneIp wi) filterW Empty_WordInterval; mtch_srcs = matching_srcs (getOneIp wi) filterW Empty_WordInterval in (map (\<lambda>d. wordinterval_element d mtch_dsts) W, map (\<lambda>s. wordinterval_element s mtch_srcs) W) in map (map fst) (groupF snd (map (\<lambda>x. (x, f x)) P)))"
fresh_2: fixes a :: "name"   and x2 :: "freeRes"   and x1 :: "pi" shows "a \<sharp> x2 \<prec> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
solve_length_generate_DB: fixes emBits :: "nat"   and M :: "bit list"   and x :: "bit list" assumes "0 < emBits"   and "length (sha1 M) + sLen + 16 \<le> emBits" shows "length (generate_DB (generate_PS emBits (length (sha1 x)))) = roundup emBits 8 * 8 - 168"
span_Int_subset: fixes A :: "'a set"   and B :: "'a set" shows "ideal (A \<inter> B) \<subseteq> ideal A \<inter> ideal B"
set_minus_filter_out: fixes xs :: "'e list"   and y :: "'e" shows "set xs - {y} = set (filter (\<lambda>x. x \<noteq> y) xs)"
C\<^sub>3: shows "\<^bold>\<forall>x. \<^bold>\<not> (\<^bold>\<forall>xa. \<^bold>\<not> (((\<^bold>\<forall>x. xa \<cdot> x \<^bold>= x \<^bold>\<leftarrow> E (xa \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> xa \<^bold>= x \<^bold>\<leftarrow> E (x \<cdot> xa))) \<^bold>\<and> E (x \<cdot> xa)))"
Arr_implies_Ide_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Ide (Cod t)"
lappend_assoc: fixes xs :: "'b llist"   and ys :: "'b llist"   and zs :: "'b llist" shows "(xs @@ ys) @@ zs = xs @@ ys @@ zs"
ex_map_conv: fixes ys :: "'f list"   and f :: "'e \<Rightarrow> 'f" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
P10: shows "A B Perp D1 D0 \<and> Coplanar A B C D1"
that: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "homeomorphism T T f g"   and "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"   and "bounded {x. \<not> (f x = x \<and> g x = x)}" shows "thesis"
e_2: shows "xs @ rs \<in> Joints (XS @ map pdevs_of_real rs)"
A: shows "\<not> \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p)"
IH: assumes "M \<in> Chains leq"   and "M \<noteq> {}" shows "lub M \<in> M"
\<tau>multithreaded_wf_axioms: shows "\<tau>multithreaded_wf final r \<tau>move"
Gromov_hyperbolic_invariant_under_quasi_isometry_explicit': fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry f" shows "Gromov_hyperbolic_subset (752 * lambda ^ 3 * (C + deltaG TYPE('d))) UNIV"
isomorphic_cards_1: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2" shows "card (uverts G\<^sub>1) = card (uverts G\<^sub>2)"
elims: fixes x :: "lit uprod"   and y :: "(lit \<times> lit) set" assumes "edges_of_clause x = y" obtains l1 :: "lit" and l2 :: "lit" where "x = Upair l1 l2" and "y = {(negate l1, l2), (negate l2, l1)}"
code: fixes z :: "gauss_int" shows "gauss_cnj z = Gauss_Int (ReZ z) (- ImZ z)"
mult_less_cancel_left_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less c zero" shows "less (times c a) (times c b) = less b a"
b_ag_group: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "Group (b_ag A)"
conseq: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P' :: "'d \<Rightarrow> 'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'d \<Rightarrow> 'f set"   and A' :: "'d \<Rightarrow> 'f set"   and P :: "'f set"   and Q :: "'f set"   and A :: "'f set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"   and "\<forall>s. s \<in> P \<longrightarrow> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"