hcju/isabelleps huggingface dataset:
propositional_atoms_def: fixes atom_ordering :: "('a \<times> 'a) set" shows "propositional_atoms atom_ordering \<equiv> (wf atom_ordering \<and> (\<forall>x y. x \<noteq> y \<longrightarrow> (x, y) \<in> atom_ordering \<or> (y, x) \<in> atom_ordering)) \<and> (\<forall>x y z. (x, y) \<in> atom_ordering \<longrightarrow> (y, z) \<in> atom_ordering \<longrightarrow> (x, z) \<in> atom_ordering) \<and> (\<forall>x y. (x, y) \<in> atom_ordering \<longrightarrow> (y, x) \<notin> atom_ordering)"
2: shows "transitive (p \<sqinter> - (1::'a))"
add_independentS_append_reduce_left: fixes As :: "'a set list"   and Bs :: "'a set list" assumes "add_independentS (As @ Bs)"   and "(0::'a) \<in> (\<Sum>X\<leftarrow>Bs. X)" shows "add_independentS As"
hn_invalidI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> hn_ctxt P x y" shows "hn_invalid P x y = true"
triangle_G: shows "G\<epsilon>o\<eta>G.map = G"
assms: fixes x :: "'a" shows "f x \<noteq> dSUCCEED"
set_to_nat_mono: fixes B :: "nat set"   and A :: "nat set" assumes "finite B"   and "A \<subseteq> B" shows "set_to_nat A \<le> set_to_nat B"
lfinite_ltake: fixes n :: "enat"   and xs :: "'a llist" shows "lfinite (ltake n xs) = (lfinite xs \<or> n < \<infinity>)"
extended_Ax_prems_empty: fixes r :: "'a sequent list \<times> 'a sequent"   and S :: "'a sequent" assumes "r \<in> Ax" shows "fst (extendRule S r) = []"
abs_reach_def: fixes \<Phi> :: "'b ltl_prop_equiv_quotient" shows "af_G_abs_opt.abs_reach \<Phi> = {foldl af_G_abs_opt.f_abs \<Phi> w |w. True}"
InvariantsVarsAfterApplyBackjump: fixes state :: "State"   and F0' :: "Literal list list"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantUniqC (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0' (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantEquivalentZL (getF state) (getM state) F0'"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)"   and "vars F0' \<subseteq> vars F0"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl" shows "let state' = applyBackjump state in InvariantVarsM (getM state') F0 Vbl \<and> InvariantVarsF (getF state') F0 Vbl \<and> InvariantVarsQ (getQ state') F0 Vbl"
\<alpha>_\<alpha>': shows "LeftDerives1 \<alpha> i r \<alpha>'"
finite_def: shows "finite \<equiv> lfp (\<lambda>p x. x = {} \<or> (\<exists>A a. x = insert a A \<and> p A))"
sublist_def: fixes xs :: "'b list"   and ys :: "'b list" shows "sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
lta_not_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "\<not> A B C CongA D E F"
assms: shows "c \<le> ESPEC E (\<lambda>r. (r \<longrightarrow> P ()) \<and> (\<not> r \<longrightarrow> E e))"
poly_add_monom: fixes p :: "'a list list"   and a :: "'a list" assumes "set p \<subseteq> carrier (K [X])"   and "a \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}" shows "UP.poly_add (UP.monom a (length p)) p = a # p"
strip_while_append_rec: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" assumes "P x" shows "strip_while P (xs @ [x]) = strip_while P xs"
uint_2_id: assumes "2 \<le> LENGTH('b)" shows "uint 2 = 2"
len_downshift: fixes P :: "'v" shows "len (downshift P) = (case len P of 0 \<Rightarrow> 0 | Suc n \<Rightarrow> n)"
set_to_map_def: fixes S :: "('b \<times> 'a) set"   and k :: "'b" shows "set_to_map S k = Eps_Opt (\<lambda>v. (k, v) \<in> S)"
Class_revI: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Orbit_Relation" shows "b \<in> orbit.Class a"
emeasure_neq_0_sets: fixes M :: "'c measure"   and A :: "'c set" assumes "emeasure M A \<noteq> 0" shows "A \<in> sets M"
B: fixes ff :: "'d list \<Rightarrow> 'd list"   and A :: "'c" shows "S (snd b ff A) = snd a ff (S A)"
dube_aux_1: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and h :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F"   and "(h, {}) \<in> set (pm_powerprod.ps X f F) \<union> set (pm_powerprod.qs ord X F)" shows "poly_deg h < max (pm_powerprod.aa X f F 1) (pm_powerprod.bb ord X F 1)"
sup_demorgan: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - x \<sqinter> - y"
simps_5: fixes f1 :: "nat \<Rightarrow> 'b"   and f2 :: "'b" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
coinduct: fixes R :: "'a stream \<Rightarrow> 'a stream \<Rightarrow> bool"   and stream :: "'a stream"   and stream' :: "'a stream" assumes "R stream stream'"   and "\<And>stream stream'. R stream stream' \<Longrightarrow> shd stream = shd stream' \<and> R (stl stream) (stl stream')" shows "stream = stream'"
oracle_decrypt1'_def: shows "oracle_decrypt1' = (\<lambda>key (bad, L) (x', c', t'). return_spmf (None, bad \<or> upf_fun (snd key) (x' @ c') = t' \<and> (x', c', t') \<notin> L \<and> length x' = prf_dlen, L))"
mset_pass\<^sub>1: fixes h :: "'b tree" shows "mset_tree (pass\<^sub>1 h) = mset_tree h"
odd_odd_even: fixes a :: "int"   and b :: "int" assumes "odd a"   and "odd b" shows "even ((a * b - 1) div 2) = even ((a - 1) div 2 + (b - 1) div 2)"
Y: shows "dist y\<^sub>0 y\<^sub>1 \<le> \<delta>"
Pmod0_principal_rev: fixes t :: "'a"   and g :: "'a" assumes "Idomain S"   and "t \<in> carrier S"   and "g \<in> carrier R"   and "\<exists>h\<in>carrier R. g = t \<cdot>\<^sub>r h" shows "P_mod R S X (S \<diamondsuit>\<^sub>p t) g"
sumdd: shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<in> carrier_mat d d"
modal_not_contain: fixes M :: "'a"   and N :: "'a"   and A :: "('b, 'a) form list"   and \<Gamma> :: "('b, 'a) form multiset" assumes "M \<noteq> N" shows "Modal M A \<notin># N \<cdot> \<Gamma>"
yEqFresh: shows "(y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X')"
assms: shows "xs \<sqsubseteq> lappend ys zs"
append_single_T_imp_tickFree: fixes t :: "'b event list"   and a :: "'b event"   and P :: "'b process" assumes "t @ [a] \<in> T P" shows "tickFree t"
QAN_Reduce: fixes s :: "trm"   and t :: "trm"   and u :: "trm" assumes "s \<leadsto> t"   and "t \<Down> u" shows "s \<Down> u"
distinct_mset_add_mset: fixes a :: "'c"   and L :: "'c multiset" shows "distinct_mset (add_mset a L) = (a \<notin># L \<and> distinct_mset L)"
red: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"
X_minus_plus: fixes a :: "'a" assumes "a \<in> carrier R" shows "X_poly_minus R a = X_poly_plus R (\<ominus> a)"
span_span: fixes A :: "'b set" shows "cspan (cspan A) = cspan A"
poincare_collinear_doubleton: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_collinear {u, v}"
simps_2: fixes oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "rec_join [oneC] = coeff_cube_to_path oneC"
p: shows "\<forall>x\<in>s - {a}. x j = 0"
focus_eq_zero_iff: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(focus X p = 0) = (p = 0)"
null_eqI: fixes n :: "'j" assumes "\<And>f. n \<cdot>\<^sub>J f = n \<and> f \<cdot>\<^sub>J n = n" shows "n = J.null"
ring_def: fixes R :: "('b, 'c) ring_scheme" shows "ring R \<equiv> abelian_group R \<and> Group.monoid R \<and> ring_axioms R"
set_child_nodes_is_l_set_child_nodes: shows "l_set_child_nodes ShadowRootClass.type_wf Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs"
is_extensional: fixes f :: "'a" assumes "\<not> \<tau>.A.arr f" shows "\<tau> f = \<tau>.B.null"
notIrr_decompose: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V"   and "\<not> IrrFinGroupRepresentation G smult V" shows "\<exists>U W. (RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V) \<and> U \<noteq> 0 \<and> U \<noteq> V \<and> (RModule (aezfun_setspan G) smult W \<and> W \<subseteq> V) \<and> W \<noteq> 0 \<and> W \<noteq> V \<and> V = U \<oplus> W"
Nml_HcompNml_4: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Nml t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "E.Src (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = E.Src u"
X: shows "Affine X = {\<Sum>b\<in>Basis. fst (y b) *\<^sub>R b..\<Sum>b\<in>Basis. snd (y b) *\<^sub>R b}"
1: shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m (store c)"
h30: shows "w_vec p qs signs = solve_for_lhs p qs subsets (M_mat signs subsets)"
sup_absorb1: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x" shows "sup x y = x"
trivial_relative_homology_group_empty: fixes X :: "'b topology"   and p :: "int"   and S :: "'b set" assumes "topspace X = {}" shows "trivial_group (relative_homology_group p X S)"
Chains_def: fixes r :: "('c \<times> 'c) set" shows "Zorn.Chains r = {C. \<forall>a\<in>C. \<forall>b\<in>C. (a, b) \<in> r \<or> (b, a) \<in> r}"
assms_2: fixes k :: "nat" assumes "k < n" shows "g k \<in> carrier_mat d d"
unfold_1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(\<box>P) \<sigma> = (P \<sigma> \<and> (\<circle>\<box>P) \<sigma>)"
mod_exp_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and e :: "nat"   and m :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "euclidean_semiring_cancel.mod_exp times one modulo b e m = modulo (power.power one times b e) m"
simps_1: fixes uu :: "('b \<times> 'b) list"   and G :: "'b graph" shows "generate_valid_topology_some [] uu G = G"
class_add_subcls: fixes D :: "char list"   and D' :: "char list"   and P :: "jvm_prog"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> D \<preceq>\<^sup>* D'"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> D \<preceq>\<^sup>* D'"
preorder_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
valid_W_invD_1: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and r :: "'g"   and fl :: "bool"   and ws :: "('e, 'f, 'g) mem_store_action list" assumes "sys_mem_store_buffers p s = mw_Mark r fl # ws"   and "valid_W_inv s" shows "fl = sys_fM s \<and> r \<in> ghost_honorary_grey (s p) \<and> sys_mem_lock s = Some p \<and> white r s \<and> filter is_mw_Mark ws = []"
inst_C: shows "C \<cdot>\<^sub>l\<^sub>s lmbd = C'"
always_different_outputs_outputs_never_equal: fixes O1 :: "vname aexp list"   and O2 :: "vname aexp list"   and s :: "vname \<Rightarrow> value option" assumes "always_different_outputs O1 O2" shows "apply_outputs O1 s \<noteq> apply_outputs O2 s"
simps_2: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and x :: "char list"   and e :: "aexp"   and Q :: "'a" shows "Step f g S (x ::= e {Q}) = x ::= e {f x e S}"
urel64_mod: fixes pp :: "uint64"   and x :: "uint64"   and x' :: "int"   and y :: "uint64"   and y' :: "int" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295"   and "urel64 x x'"   and "urel64 y y'" shows "urel64 (if y = 0 then x else 0) (if y' = 0 then x' else 0)"
Un_ac_3: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B = B \<union> A"
mat1: fixes env :: "v sem_env"   and s1 :: "'c state"   and e :: "exp0"   and s2 :: "'c state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list"   and e' :: "exp0"   and env' :: "(char list \<times> v) list"   and bv :: "'c state \<times> (v, v) result" assumes "Big_Step_Unclocked.evaluate env s1 e (s2, Rval v1)"   and "match_result env s2 v1 pes Bindv = Rval (e', env')"   and "Big_Step_Unclocked.evaluate (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s2 e' bv" shows "Big_Step_Unclocked.evaluate env s1 (Mat e pes) bv"
interval_image_affinity_interval: fixes m :: "real"   and c :: "'a"   and a :: "'a"   and b :: "'a" shows "\<exists>u v. (\<lambda>x. m *\<^sub>R x + c) ` cbox a b = cbox u v"
subst_apply_extstrand_nil: fixes \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "[] \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> = []"
assms: shows "\<langle>c \<and> t\<rangle> \<le> n"
llast_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "llast (xs $ ys) = (if lnull ys then llast xs else if lfinite xs then llast ys else undefined)"
simps_5: fixes p :: "('c, 'b) fm" shows "nominals (\<^bold>\<diamond> p) = nominals p"
le_dbm_le: fixes a :: "'c"   and b :: "'c" assumes "Le a \<preceq> Le b" shows "a \<le> b"
fold_graph_And\<^sub>n: fixes \<Phi> :: "'b ltln set" assumes "finite \<Phi>" shows "fold_graph And_ltln true\<^sub>n \<Phi> (And\<^sub>n \<Phi>)"
bad_outputs: fixes t1 :: "transition"   and r :: "nat \<Rightarrow>f value option"   and t2 :: "transition" assumes "\<exists>i. can_take_transition t1 i r \<and> evaluate_outputs t1 i r \<noteq> evaluate_outputs t2 i r" shows "\<not> subsumes t2 r t1"
h2: shows "vars (liftPoly i (length I) p) \<inter> {i..<i + length I} = {}"
lhs: shows "id_func AA \<^bsub>\<a>\<^esub> (g \<bullet> f) = g \<bullet> f"
derivable_def: shows "derivable \<equiv> \<lambda>R. {(xa, x). derivablep (\<lambda>x. x \<in> R) xa x}"
welldefined_subsets: shows "all_list_constr subsets (length qs)"
inj_on_chamber: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C" shows "inj_on f C"
is_span: shows "Maps.span Cod'.leg0 Cod'.leg1"
cong: fixes g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "\<And>a. g a = h a" shows "G g = G h"
valid_decomp_concat: fixes pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list"   and X :: "'x set" assumes "\<And>ps. ps \<in> set pss \<Longrightarrow> valid_decomp X ps" shows "valid_decomp X (concat pss)"
singleton_def: fixes default :: "unit \<Rightarrow> 'b"   and A :: "'b Predicate.pred" shows "Predicate.singleton default A = (if \<exists>!x. pred.eval A x then THE x. pred.eval A x else default ())"
hexp_def: fixes B :: "hf"   and C :: "hf" shows "hexp B C = HCollect (hfun B C) (HPow (B * C))"
that: fixes t :: "'a"   and u :: "'t" assumes "t \<in> keys p"   and "u \<in> keys q"   and "v = t \<oplus> u" shows "thesis"
simps_2: fixes xs :: "('b, 'c) Combinators list" shows "net_list_aux (DenyAll # xs) = net_list_aux xs"
pStates_finite: fixes prog :: "program" shows "finite (Collect (pState_inv prog))"
Q: shows "Q \<in> subgroups_of_size (p ^ a)"
take_bit_int_def: fixes n :: "nat"   and k :: "int" shows "take_bit n k = k mod 2 ^ n"
fls_lr_inverse_zero_2: fixes y :: "'a" shows "fls_right_inverse 0 y = fls_const y"
assms_1: shows "\<forall>twoC\<in>{circle_cube}. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
Collect_subset: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
minus_closed: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<ominus>\<^bsub>G\<^esub> y \<in> carrier G"
simps_1: fixes j :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_a 0 j A = A"
select_convs_3: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, \<dots> = more\<rparr>\<^esub> = zero"
strict_implies_order: fixes a :: "'d"   and b :: "'d" assumes "a < b" shows "a \<le> b"
w: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> Y" shows "wave \<Gamma> f"
rat: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < i" shows "rat_of_int (d\<mu> fs i j) = rat_of_int (d fs (Suc j)) * fs.gs.\<mu> i j"
star_denest: fixes x :: "'b"   and y :: "'b" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> \<cdot> y\<^sup>\<star>)\<^sup>\<star>"
drop_len: fixes u :: "'b list"   and w :: "'b list"   and v :: "'b list" shows "\<^bold>|u \<cdot> w\<^bold>| \<le> \<^bold>|u \<cdot> v \<cdot> w\<^bold>|"
map_sel_1: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d sctn" shows "normal (map_sctn f a) = f (normal a)"
function_ring_not_car: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "a \<notin> S"   and "f \<in> carrier (function_ring S R)" shows "f a = undefined"
monomial_functions: fixes c :: "nat \<Rightarrow> int"   and n :: "nat" assumes "c \<in> carrier Zp" shows "monomial c n \<in> carrier (function_ring (carrier Zp) Zp)"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b err"   and b :: "'c err" assumes "rel_err R a b" obtains "a = Err" and "b = Err"   | x :: "'b" and y :: "'c" where "a = OK x" and "b = OK y" and "R x y"
equality: fixes r :: "('d, 'f, 'e, 'g, 'h) galcon_scheme"   and r' :: "('d, 'f, 'e, 'g, 'h) galcon_scheme" assumes "\<X>\<^bsub>r\<^esub> = \<X>\<^bsub>r'\<^esub>"   and "\<Y>\<^bsub>r\<^esub> = \<Y>\<^bsub>r'\<^esub>"   and "\<pi>\<^sup>*\<^bsub>r\<^esub> = \<pi>\<^sup>*\<^bsub>r'\<^esub>"   and "\<pi>\<^sub>*\<^bsub>r\<^esub> = \<pi>\<^sub>*\<^bsub>r'\<^esub>"   and "galcon.more r = galcon.more r'" shows "r = r'"
linear_compose_add: fixes f :: "('a \<Rightarrow> real) \<Rightarrow> real"   and g :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> f x \<in> UNIV"   and "\<forall>x. x \<in> diff_fun_space \<longrightarrow> g x \<in> UNIV"   and "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) f"   and "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) g" shows "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) (\<lambda>x. f x + g x)"
monom_mult_Sum_any_left: fixes f :: "'b \<Rightarrow> 'c"   and t :: "'d"   and p :: "'d \<Rightarrow>\<^sub>0 'c" assumes "finite {c. f c \<noteq> (0::'c)}" shows "MPoly_Type_Class.punit.monom_mult (Sum_any f) t p = (\<Sum>c. MPoly_Type_Class.punit.monom_mult (f c) t p)"
empt_invariant: shows "rquot_invariant A (rquot_empt A)"
rep_eq: fixes x :: "integer" shows "msb x = msb (int_of_integer x)"
trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "bob x y"   and "bob y z"   and "x \<in> B" shows "bob x z"
path_split_set': fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v"   and W :: "'v set" assumes "is_path v p v'"   and "v' \<in> W" obtains p1 :: "('v \<times> 'w \<times> 'v) list" and p2 :: "('v \<times> 'w \<times> 'v) list" and u :: "'v" where "p = p1 @ p2" and "is_path v p1 u" and "is_path u p2 v'" and "int_vertices p1 \<subseteq> - W" and "u \<in> W"
timpl_closure'_inv: fixes s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" assumes "(s, t) \<in> timpl_closure' TI" shows "(\<exists>x. s = Var x \<and> t = Var x) \<or> (\<exists>f g S T. s = Fun f S \<and> t = Fun g T \<and> length S = length T)"
UnOp: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and Te :: "ty"   and unop :: "unop"   and T :: "ty" assumes "E,dt\<Turnstile>e\<Colon>-Te"   and "wt_unop unop Te"   and "T = PrimT (unop_type unop)" shows "E,dt\<Turnstile>UnOp unop e\<Colon>-T"
index_same_def: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd" shows "index_same index key \<equiv> \<forall>x y n mi ma. key x \<in> {mi..ma} \<longrightarrow> x \<noteq> y \<longrightarrow> key x = key y \<longrightarrow> index key x n mi ma = index key y n mi ma"
dep_in_imp_supset_circuit_in: fixes X :: "'a set" assumes "X \<subseteq> \<E>"   and "\<not> indep_in \<E> X" shows "\<exists>C. circuit_in \<E> C \<and> C \<subseteq> X"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
infinite_super: fixes S :: "'d set"   and T :: "'d set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
one_add_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
simps_1: fixes t :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle>) = {t}"
Timpl_rule: fixes s :: "bdd" assumes "bdd_sane s" shows "ospec (tmi' s) (\<lambda>(ni, s'). (ni, Trueif) \<in> Rmi s' \<and> bdd_sane s' \<and> mi.les s s')"
processed_Nil: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list" shows "processed (a, b) xs [] = (a \<in> set xs \<and> b \<in> set xs)"
subst: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "subst \<phi> t x \<in> fmla"
replace_comp_cls_val_process0: fixes Mkt :: "('c, 'd) discrete_market"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>w. prices Mkt x 0 w = cls_val_process Mkt pf2 0 w"   and "portfolio pf1"   and "portfolio pf2" shows "\<forall>w. cls_val_process Mkt (qty_replace_comp pf1 x pf2) 0 w = cls_val_process Mkt pf1 0 w"
eq_onp_def: fixes R :: "'n \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
FL_valid_Conj: fixes xset :: "('a, 'pred, 'act, 'effect) formula set['a]"   and P :: "'state" assumes "finite (supp xset)" shows "P \<Turnstile> Conj xset = (\<forall>x\<in>set_bset xset. P \<Turnstile> x)"
zip_commute: fixes xs :: "'g list"   and ys :: "'h list" shows "zip xs ys = map2 (\<lambda>x y. (y, x)) ys xs"
assms: fixes i :: "'a" shows "ket i \<bullet>\<^sub>C \<psi> = ket i \<bullet>\<^sub>C \<phi>"
cgenideal_eq_rcos: fixes R :: "('e, 'f) ring_scheme"   and i :: "'e" assumes "cring R" shows "PIdl\<^bsub>R\<^esub> i = carrier R #>\<^bsub>R\<^esub> i"