hcju/isabelleps huggingface dataset:
interval_bounds'_1: fixes a :: "'c"   and b :: "'c" assumes "cbox a b \<noteq> {}" shows "interval_upperbound (cbox a b) = b"
phiDefs_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and n :: "'node" shows "uninst_code.ssa.phiDefs p g n \<equiv> {v. (n, v) \<in> dom (Mapping.lookup (p g))}"
graph_single: fixes a :: "'b"   and b :: "'c"   and c :: "'c" shows "graph (LG {(a, b, c)} {b, c})"
axioms_1: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" assumes "disc_filtr_prob_space M F" shows "prob_space M"
bind_rzero: fixes x :: "'f option" shows "x \<bind> Map.empty = None"
int_shiftr_numeral_Suc0_1: shows "1 >> Suc 0 = 0"
preal_le_def: fixes r :: "preal"   and s :: "preal" shows "r \<le> s \<equiv> Rep_preal r \<subseteq> Rep_preal s"
sum_unfold_sum_mset: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f A = comm_monoid_add.sum_mset plus zero (image_mset f (mset_set A))"
Reals_add_cancel: fixes x :: "'a"   and y :: "'a" assumes "x + y \<in> \<real>"   and "y \<in> \<real>" shows "x \<in> \<real>"
gval_foldr_true: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "(gval (foldr gAnd G (Bc True)) s = true) = (\<forall>g\<in>set G. gval g s = true)"
ball_one_point2: fixes A :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>A. a = x \<longrightarrow> P x) = (a \<in> A \<longrightarrow> P a)"
absorb2: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" assumes "b \<preceq> a" shows "linordered_monoid.min a b = b"
valid_def: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "G \<Turnstile> c : A  = ( \<Turnstile> G  \<longrightarrow>  \<Turnstile> c : A )"
closed: fixes x :: "'a"   and y :: "'a" assumes "x .= y"   and "x \<in> A"   and "y \<in> carrier S" shows "y \<in> A"
mset_ran_single: fixes a :: "int \<Rightarrow> 'b"   and i :: "int" shows "mset_ran a {i} = {#a i#}"
weak_transition_not_tau_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and \<alpha> :: "'c"   and P :: "'a"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "\<alpha> \<noteq> \<tau>" shows "weak_nominal_ts.weak_transition transition \<tau> P \<alpha> P' = weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'"
zero_closed: shows "\<zero>\<^bsub>R Quot I\<^esub> \<in> carrier (R Quot I)"
lit_numeral: fixes n :: "num" shows "\<guillemotleft>numeral n\<guillemotright> = numeral n"
A: shows "{#a#} + {#b#} \<subseteq># A + B"
fim: shows "f ` topspace (nsphere p) \<noteq> topspace (nsphere p)"
Un_insert_right: fixes A :: "'e set"   and a :: "'e"   and B :: "'e set" shows "A \<union> insert a B = insert a (A \<union> B)"
independent_vnm_alt_def: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" shows "independent_vnm C P = (\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..<1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r)"
ag_idmap_def: fixes A :: "('c, 'd) carrier_scheme" shows "aI\<^bsub>A\<^esub> = (\<lambda>x\<in>carrier A. x)"
lookup_operator_works_2: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "lookup_action (abs_ast_operator (name, pres, effs, layer)) = Some (name', pres', effs', layer')" shows "pres = pres'"
gallery_chamber_system: fixes Cs :: "'a set list" assumes "folding_g.gallery Cs" shows "order.greater_eq folding_g.\<C> (set Cs)"
meetE: fixes b :: "'b"   and c :: "'b"   and A :: "('b \<times> 'b) set" assumes "(b, c) \<in> A\<^sup>\<up>" obtains a :: "'b" where "(a, b) \<in> A\<^sup>*" and "(a, c) \<in> A\<^sup>*"
mem_val_w32_def: fixes asi :: "word8"   and addr :: "word36"   and state :: "'b sparc_state" shows "mem_val_w32 asi addr state \<equiv> let addr' = addr AND 68719476732; addr0 = addr'; addr1 = addr' + 1; addr2 = addr' + 2; addr3 = addr' + 3; r0 = mem_val_alt asi addr0 state; r1 = mem_val_alt asi addr1 state; r2 = mem_val_alt asi addr2 state; r3 = mem_val_alt asi addr3 state in if r0 = None \<or> r1 = None \<or> r2 = None \<or> r3 = None then None else let byte0 = case r0 of Some v \<Rightarrow> v; byte1 = case r1 of Some v \<Rightarrow> v; byte2 = case r2 of Some v \<Rightarrow> v; byte3 = case r3 of Some v \<Rightarrow> v in Some ((((ucast byte0 << 24) OR (ucast byte1 << 16)) OR (ucast byte2 << 8)) OR ucast byte3)"
assms_3: shows "has_least_fixpoint (g \<circ> f)"
Rep_upset_inject: fixes x :: "'a upset"   and y :: "'a upset" shows "(Rep_upset x = Rep_upset y) = (x = y)"
pfp_pfp: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and x :: "'b" assumes "pfp f x0 = Some x" shows "f x \<sqsubseteq> x"
simps: fixes x :: "nat"   and y :: "nat" shows "uminus_int_raw (x, y) = (y, x)"
Sum_any_zeroI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x. f x = (0::'c)" shows "Sum_any f = (0::'c)"
measurable_dens_ctxt_fun_upd: fixes f :: "'a \<Rightarrow> nat \<Rightarrow> val"   and N :: "'a measure"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and V :: "nat set"   and x :: "nat"   and g :: "'a \<Rightarrow> val" assumes "f \<in> N \<rightarrow>\<^sub>M state_measure V' \<Gamma>"   and "V = V' \<union> {x}"   and "g \<in> N \<rightarrow>\<^sub>M stock_measure (\<Gamma> x)" shows "(\<lambda>\<omega>. (f \<omega>)(x := g \<omega>)) \<in> N \<rightarrow>\<^sub>M state_measure V \<Gamma>"
proj_mult_scalar: fixes k :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "proj_poly k (p \<odot> q) = p * proj_poly k q"
cnt: fixes fi :: "int poly"   and i :: "nat" assumes "(fi, i) \<in> set psi" shows "primitive fi"
weak_partial_order_bottom_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_partial_order_bottom_axioms L \<equiv> \<exists>x. least L x (carrier L)"
differentiable_inverse: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "f differentiable at x within s"   and "f x \<noteq> (0::'b)" shows "(\<lambda>x. inverse (f x)) differentiable at x within s"
all_chains_no_call_upd_r: fixes c :: "char list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and x :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "all_chains (no_call_to c) \<Gamma> rs" shows "\<Gamma>(c \<mapsto> x),\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t = \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r t"
apf_def: fixes x :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'c" shows "x \<diamondop> y = (\<lambda>z. x z (y z))"
isomorphic_objects_are_equal: fixes a :: "'d"   and b :: "'d" assumes "D.obj a"   and "D.obj b"   and "D.isomorphic a b" shows "a = b"
\<R>_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "closed_glob_consistent (\<R> \<Gamma>')"
floor_correct: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and floor :: "'c \<Rightarrow> int"   and x :: "'c" assumes "class.floor_ceiling divide inverse times one plus zero minus uminus less_eq less sgn abs floor" shows "less_eq (ring_1.of_int one plus zero minus (floor x)) x \<and> less x (ring_1.of_int one plus zero minus (floor x + 1))"
inv_allD: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "??.Jordan_Normal_Form_Existence.inv_all n p A"   and "i < n"   and "j < n" shows "p A i j"
Lim_ident_at: fixes x :: "'c"   and s :: "'c set" assumes "at x within s \<noteq> bot" shows "netlimit (at x within s) = x"
zip_take_triv: fixes n :: "nat"   and ys :: "'b list"   and xs :: "'c list" assumes "n = length ys" shows "zip (take n xs) ys = zip xs ys"
mindeg_def: fixes A :: "(('e \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'f) set" shows "mindeg A = Min (poly_deg ` A)"
aligned_shift: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "x < 2 ^ n"   and "is_aligned y n"   and "n \<le> LENGTH('b)" shows "x + y >> n = y >> n"
msg_ok_def: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" shows "msg_ok \<sigma> m \<equiv> case m of Pkt num' sid' \<Rightarrow> num' \<le> no (\<sigma> sid') | Newpkt nat1 nat2 \<Rightarrow> True"
g_ball_def: fixes m :: "('b, 'c) assoc_list"   and P :: "'b \<times> 'c \<Rightarrow> bool" shows "lm_basic.g_ball m P \<equiv> lm_basic.iteratei m id (\<lambda>kv \<sigma>. P kv) True"
ESPEC_rule_iff: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool"   and \<Phi>' :: "'c \<Rightarrow> bool"   and \<Psi>' :: "'d \<Rightarrow> bool" shows "(ESPEC \<Phi> \<Psi> \<le> ESPEC \<Phi>' \<Psi>') = ((\<forall>e. \<Phi> e \<longrightarrow> \<Phi>' e) \<and> (\<forall>x. \<Psi> x \<longrightarrow> \<Psi>' x))"
scaleR_nonpos_nonpos: fixes a :: "real"   and b :: "'a" assumes "a \<le> 0"   and "b \<le> (0::'a)" shows "(0::'a) \<le> a *\<^sub>R b"
countable: fixes x :: "'a" assumes "x \<in> A" shows "countable (B x)"
monom_mult_pprod_def: shows "monom_mult_pprod \<equiv> pprod.monom_mult"
induct: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and x1 :: "char list"   and x2 :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and Pa :: "char list \<Rightarrow> (char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option) \<Rightarrow> bool" assumes "P \<turnstile> x1 sees_methods x2"   and "\<And>D fs ms Mm. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms\<rbrakk> \<Longrightarrow> Pa Object Mm"   and "\<And>C D fs ms Mm Mm'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D sees_methods Mm; Pa D Mm; Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)\<rbrakk> \<Longrightarrow> Pa C Mm'" shows "Pa x1 x2"
seq_Inf_distrib: fixes D :: "'a::refinement_lattice set"   and c :: "'a::refinement_lattice" assumes "(D::'a::refinement_lattice set) \<noteq> {}" shows "(c::'a::refinement_lattice) ; \<Sqinter> (D::'a::refinement_lattice set) = \<Sqinter> ((;) c ` D)"
next_plane0_finalVertex_mono: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and u :: "nat" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "u \<in> \<V> g"   and "finalVertex g u" shows "finalVertex g' u"
z_def: shows "z = (\<Sum>i\<in>Basis. (real (q (b' i)) / real p) *\<^sub>R i)"
max_correct_2: fixes s :: "'s"   and x :: "'u"   and P :: "'u \<Rightarrow> bool" assumes "invar s"   and "x \<in> \<alpha> s"   and "P x" shows "x \<le> the (max s P)"
frag_of_nonzero: fixes a :: "'c" shows "frag_of a \<noteq> 0"
sup_commute: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
similarity_scale_1: fixes k :: "complex"   and I :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0" shows "similarity (k *\<^sub>s\<^sub>m I) M = similarity I M"
base_pair2: fixes x :: "state \<Rightarrow> 'a"   and y :: "state \<Rightarrow> 'b" assumes "basevars (x, y)" shows "basevars y"
g'_nz: fixes z :: "complex" assumes "ereal (cmod z) < min r (fps_conv_radius g)" shows "eval_fps g' z \<noteq> 0"
fold_coeffs_def: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'b poly" shows "fold_coeffs f p = foldr f (coeffs p)"
P'_altdef: shows "P' = (\<Prod>X\<in>Pow {..<n} - {{}}. [:- root' X, 1:])"
one_closed: shows "\<one>\<^bsub>B\<^esub> \<in> K"
simps_1: fixes \<mu> :: "'a" shows "Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
cts_wp_loopstep: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "bd_cts (wp body)" shows "bd_cts_tr (\<lambda>x. wp (body ;; Embed x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
IH: assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X"
sfirst_Stream: fixes P :: "'a stream \<Rightarrow> bool"   and s :: "'a"   and x :: "'a stream" shows "sfirst P (s ## x) = (if P (s ## x) then 0 else eSuc (sfirst P x))"
pw_THROW_2: fixes e :: "'c"   and f :: "'c" shows "inres (THROW e) (Inl f) = (f = e)"
wf_graph_generate_valid_topology: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list" assumes "wf_graph G" shows "wf_graph (generate_valid_topology M G)"
FO_resolution_prover_axioms: shows "FO_resolution_prover S (\<cdot>a) id_subst (\<odot>) renamings_apart atm_of_atms mgu less_atm"
cong3_diff: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "A B C Cong3 A' B' C'" shows "A' \<noteq> B'"
dist_triangle_add: fixes x :: "'a"   and y :: "'a"   and x' :: "'a"   and y' :: "'a" shows "dist (x + y) (x' + y') \<le> dist x x' + dist y y'"
ltake_ldrop_id: fixes x :: "'b llist"   and i :: "nat" shows "x \<down> i @@ x \<up> i = x"
unreachable_Null_simp: fixes s :: "Store"   and l :: "Location" shows "\<not> s\<turnstile> l reachable_from nullV"
snd_inf: fixes x :: "'b \<times> 'a"   and y :: "'b \<times> 'a" shows "snd (x \<sqinter> y) = snd x \<sqinter> snd y"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'b" shows "mono f = (\<forall>n. f n \<le> f (Suc n))"
two_comparisons_into_case_order_5: fixes x :: "'b"   and y :: "'b"   and Q :: "'d"   and P :: "'d"   and R :: "'d" shows "(if c1.lt x y then Q else if c1.le x y then P else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
moebius_mu_formula_int: fixes n :: "nat" shows "moebius_mu n = dirichlet_inverse (\<lambda>_. 1) 1 n"
finite_UNIV_fun_def: shows "finite_UNIV = Phantom('a \<Rightarrow> 'b) (let cb = of_phantom card_UNIV_class.card_UNIV in cb = 1 \<or> of_phantom finite_UNIV \<and> cb \<noteq> 0)"
rev_exhaust: fixes xs :: "'f list" obtains "xs = []"   | ys :: "'f list" and y :: "'f" where "xs = ys @ [y]"
post_meas0_bell01_snd: shows "post_meas0 2 |\<beta>\<^sub>0\<^sub>1\<rangle> 1 = |unit_vec 4 2\<rangle>"
is_processT8_S: fixes s :: "'b event list"   and P :: "'b process"   and X :: "'b event set" assumes "s \<in> D P" shows "(s, X) \<in> F P"
prj_simps_10: shows "C.arr gfg.Prj\<^sub>1"
terminal_def: fixes tllist :: "('g, 'f) tllist" shows "terminal tllist = (case tllist of TNil x1 \<Rightarrow> x1 | TCons _dummy_ xs \<Rightarrow> terminal0 xs)"
fmfilter_alt_defs_4: fixes A :: "'c set" shows "fmrestrict_set A = (\<lhd>) A"
class_les_def: fixes sub :: "('a \<times> 'a) set"   and c1 :: "'a"   and c2 :: "'a" shows "class_les sub c1 c2 = (class_leq sub c1 c2 \<and> \<not> class_leq sub c2 c1)"
1: shows "\<Theta>,\<Gamma> \<turnstile> subst_bv (Fv x \<tau>) (strip_all_single_body B)"
f_ind: fixes anything :: "nat list" assumes "rec_exec f (map (\<lambda>g. rec_exec g xs) gs) = rec_exec (Cn (length xs) f gs) xs" shows "{\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (ffp - far) @ anything} fap {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (ffp - Suc far) @ anything}"
Sup_power_expand: fixes P :: "nat \<Rightarrow> 'b" shows "P 0 \<sqinter> (\<Sqinter>i. P (i + 1)) = \<Sqinter>range P"
parallel_reduction_subst_inner: fixes M :: "'a trm"   and M' :: "'a trm"   and X :: "'a trm"   and z :: "'a" assumes "M >> M'" shows "X[z ::= M] >> X[z ::= M']"
learn_bcI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. \<psi> (the (s (f \<triangleright> n))) = f" shows "learn_bc \<psi> U s"
thetaWhileW0_sym: shows "sym thetaWhileW0"
run_state_get: fixes s :: "'b" shows "run_state get s = (s, s)"
id_prs: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g" assumes "Quotient3 R Abs Rep" shows "(Rep ---> Abs) id = id"
fw_shortest: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "linordered_monoid.cycle_free m n"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "linordered_monoid.fw m n n n n i j \<preceq> dbm_add (linordered_monoid.fw m n n n n i k) (linordered_monoid.fw m n n n n k j)"
that_2: fixes ys :: "'a fset"   and x :: "'a" assumes "x |\<notin>| ys"   and "xs = insert_fset x ys" shows "thesis"
wf_norm_rderivs: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and k :: "nat" assumes "local.wf idx \<phi>" shows "local.wf idx (((norm \<circ> rderiv (zero idx)) ^^ k) \<phi>)"
ccSup_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'c set"   and A :: "'c set" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable B"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
lookup_dflt_eq_lookup_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "lookup_dflt xs = lookup_pair x xs"
S5n_jAction_eq: fixes M :: "('a, 'p, 'b) KripkeStructure"   and w :: "'b"   and w' :: "'b"   and a :: "'a" assumes "S5n M"   and "(w, w') \<in> relations M a" shows "jAction M w a = jAction M w' a"
bdd: fixes n :: "nat"   and x :: "real" shows "\<bar>f n x\<bar> \<le> M"
piecewise_C1_differentiable_on_subset: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set"   and T :: "real set" assumes "f piecewise_C1_differentiable_on S"   and "T \<subseteq> S" shows "f piecewise_C1_differentiable_on T"
z: shows "(P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y)"
less_1_prodinf: fixes f :: "nat \<Rightarrow> real" assumes "convergent_prod f"   and "\<And>n. 1 < f n" shows "1 < prodinf f"
assms_1: shows "subst_domain \<sigma> \<inter> range_vars \<theta> = {}"
units_of_one: fixes G :: "('c, 'd) monoid_scheme" shows "\<one>\<^bsub>units_of G\<^esub> = \<one>\<^bsub>G\<^esub>"
wfE_pf: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf R"   and "A \<subseteq> R `` A" shows "A = {}"
less_option_None_Some: fixes x :: "'a" shows "None < Some x"
map_G_comp: fixes l1 :: "'p \<Rightarrow> 'j"   and l2 :: "'q \<Rightarrow> 'k"   and co1 :: "'r \<Rightarrow> 'l"   and co2 :: "'s \<Rightarrow> 'm"   and contra1 :: "'n \<Rightarrow> 't"   and contra2 :: "'o \<Rightarrow> 'u"   and l1' :: "'b \<Rightarrow> 'p"   and l2' :: "'d \<Rightarrow> 'q"   and co1' :: "'e \<Rightarrow> 'r"   and co2' :: "'f \<Rightarrow> 's"   and contra1' :: "'t \<Rightarrow> 'g"   and contra2' :: "'u \<Rightarrow> 'h" shows "map_G l1 l2 co1 co2 contra1 contra2 \<circ> map_G l1' l2' co1' co2' contra1' contra2' = map_G (l1 \<circ> l1') (l2 \<circ> l2') (co1 \<circ> co1') (co2 \<circ> co2') (contra1' \<circ> contra1) (contra2' \<circ> contra2)"
wf_tuple_restrict: fixes n :: "nat"   and B :: "nat set"   and v :: "'b option list"   and A :: "nat set"   and C :: "nat set" assumes "wf_tuple n B v"   and "A \<inter> B = C" shows "wf_tuple n C (restrict A v)"
almost_full_on_UNIV_nat: shows "almost_full_on (\<le>) UNIV"
basis_reduction_main: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list"   and fs' :: "int vec list" assumes "LLL_invariant upw i fs"   and "basis_reduction_main (upw, i, fs) = fs'" shows "LLL_invariant True m fs'"
eval_cases_6: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>Var V,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains l :: "char list \<Rightarrow> val option" and v :: "val" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" where "s = (h, l)" and "e' = Val v" and "s' = (h, l)" and "l V = \<lfloor>v\<rfloor>"
tortoise_hare: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" assumes "properties f x0 lambda mu" shows "\<lbrace>\<lambda>s. True\<rbrace> fx0.tortoise_hare f x0 \<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> l s = lambda \<and> m s = mu\<rbrace>"
holfm: fixes m :: "nat" shows "f ^^ m holomorphic_on s"
eval_trm_graph_aux_def: shows "eval_trm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>eval_trm_aux e. x1 = (e, zer) \<and> x2 = 0) \<or> (\<exists>eval_trm_aux e k. x1 = (e, Var k) \<and> x2 = finfun_apply e k) \<or> (\<exists>eval_trm_aux e t. x1 = (e, suc t) \<and> x2 = Suc (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t)) \<and> p (e, t) (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t))) \<or> (\<exists>eval_trm_aux e t u. x1 = (e, pls t u) \<and> x2 = ((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t) + ((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, u) \<and> p (e, t) (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t)) \<and> p (e, u) (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, u))) \<or> (\<exists>eval_trm_aux e t u. x1 = (e, tms t u) \<and> x2 = ((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t) * ((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, u) \<and> p (e, t) (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, t)) \<and> p (e, u) (((\<lambda>uu_. uu_) \<circ> eval_trm_aux) (e, u))))"
f_integrable: shows "set_integrable lborel (einterval a b) f"
inter_add_left1: fixes x :: "'a"   and N :: "'a multiset"   and M :: "'a multiset" assumes "x \<notin># N" shows "add_mset x M \<inter># N = M \<inter># N"
properfactor_def: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" shows "Divisibility.properfactor G a b = (a divides\<^bsub>G\<^esub> b \<and> \<not> b divides\<^bsub>G\<^esub> a)"
Card_order_cmax: fixes r :: "('b \<times> 'b) set"   and s :: "('c \<times> 'c) set" assumes "Card_order r"   and "Card_order s" shows "Card_order (cmax r s)"