hcju/isabelleps huggingface dataset:
root_list_ub: fixes ps :: "real poly list"   and a :: "real" assumes "0 \<notin> set ps" obtains ub :: "real" where "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub" and "\<forall>x\<ge>ub. \<forall>p\<in>set ps. sgn (poly p x) = sgn_pos_inf p" and "a < ub"
simps_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and x :: "'a"   and y :: "'a" assumes "flow_network \<Delta> g" shows "flow_network.h_plus \<Delta> g (Suc i) (x, y) = (if flow_network.enum_v \<Delta> (Suc i) = x \<and> d_OUT (flow_network.h_plus \<Delta> g i) x < d_IN (flow_network.h_plus \<Delta> g i) x then let total = d_IN (flow_network.h_plus \<Delta> g i) x - d_OUT (flow_network.h_plus \<Delta> g i) x; share = g (x, y) - flow_network.h_plus \<Delta> g i (x, y); shares = d_OUT g x - d_OUT (flow_network.h_plus \<Delta> g i) x in flow_network.h_plus \<Delta> g i (x, y) + share * total / shares else flow_network.h_plus \<Delta> g i (x, y))"
inext_nth_Suc_mono2: fixes I :: "nat set"   and n :: "nat" assumes "\<exists>x\<in>I. I \<rightarrow> n < x" shows "I \<rightarrow> n < I \<rightarrow> Suc n"
Arg2pi_eq_Im_Ln: fixes z :: "complex" assumes "0 \<le> Im z"   and "0 < Re z" shows "Arg2pi z = Im (Ln z)"
divide_1: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oslash> \<one> = x"
expr: shows "\<exists>pr. price_structure der matur \<pi> pr \<and> (\<forall>x. x \<notin> stocks Mkt \<longrightarrow> (\<forall>Mkt2 p. coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr \<and> portfolio p \<and> support_set p \<subseteq> stocks Mkt \<union> {x} \<longrightarrow> \<not> arbitrage_process Mkt2 p))"
trivial_integer_mod_group: fixes n :: "nat" shows "trivial_group (integer_mod_group n) = (n = 1)"
op_list_append_elem_def: fixes s :: "'b list"   and x :: "'b" shows "op_list_append_elem s x \<equiv> s @ [x]"
bij_inv_eq_iff: fixes p :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c" assumes "bij p" shows "(x = inv p y) = (p x = y)"
dg_Rel_is_arrE: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B" obtains "arr_Rel \<alpha> S" and "S\<lparr>ArrDom\<rparr> = A" and "S\<lparr>ArrCod\<rparr> = B"
bhv_hn_ex: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and tid :: "'a"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "honest tid"   and "\<exists>n'\<ge>n. active tid (t n')"   and "\<exists>n'<n. active tid (t n')"   and "\<exists>b\<in>pin (cmp tid (t (dynamic_component.latestAct active tid t n))). length (bc (cmp tid (t (dynamic_component.latestAct active tid t n)))) < length b" shows "\<not> mining (cmp tid (t (dynamic_component.nxtAct active tid t n))) \<and> bc (cmp tid (t (dynamic_component.nxtAct active tid t n))) = MAX (pin (cmp tid (t (dynamic_component.latestAct active tid t n)))) \<or> mining (cmp tid (t (dynamic_component.nxtAct active tid t n))) \<and> (\<exists>b. bc (cmp tid (t (dynamic_component.nxtAct active tid t n))) = MAX (pin (cmp tid (t (dynamic_component.latestAct active tid t n)))) @ [b])"
card_2_iff': fixes S :: "'c set" shows "(card S = 2) = (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
isomorphic_transitive: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "VoV.isomorphic f g"   and "VoV.isomorphic g h" shows "VoV.isomorphic f h"
INF_partition_nat3: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'b::refinement_lattice"   and i :: "nat" shows "\<Sqinter> range ((f::nat \<Rightarrow> nat \<Rightarrow> 'b::refinement_lattice) (i::nat)) = \<Sqinter> (f i ` {j::nat. i = j}) \<sqinter> \<Sqinter> (f i ` {j::nat. i < j}) \<sqinter> \<Sqinter> (f i ` {j::nat. j < i})"
val_def: fixes tm :: "'a tm" shows "tm.val tm = (case tm of TM x1 x2 \<Rightarrow> x1)"
freshEnv_getEnv: fixes ys :: "'b"   and y :: "'c"   and rho :: "'b \<Rightarrow> 'c \<Rightarrow> ('d, 'e, 'b, 'c, 'f) qTerm set option" assumes "freshEnv ys y rho" shows "rho ys y = None"
image_replicate_mset: fixes f :: "'l \<Rightarrow> 'k"   and n :: "nat"   and a :: "'l" shows "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
finite_subst_domain_subst_of: fixes xs :: "('c \<times> ('d, 'c) Term.term) list" shows "finite (subst_domain (subst_of xs))"
liftAll_def: fixes phi :: "'a \<Rightarrow> bool"   and inp :: "'b \<Rightarrow> 'a option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
subsJustified': fixes ats :: "(signs \<times> predicate \<times> vbl list) list"   and n :: "nat"   and f :: "formula"   and list :: "(nat \<times> formula) list" shows "\<not> SATAxiom (sequent (ats, (n, f) # list)) \<longrightarrow> \<not> terminal subs (ats, (n, f) # list) \<longrightarrow> (\<forall>sigma\<in>subs (ats, (n, f) # list). sequent sigma \<in> deductions CutFreePC) \<longrightarrow> sequent (ats, (n, f) # list) \<in> deductions CutFreePC"
matches_dest_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and m' :: "'b match_expr"   and a :: "action"   and p :: "'c" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m a p" shows "False"
distinct_1: fixes x2 :: "'b" shows "NoMsg \<noteq> Msg x2"
gFreshBinp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme"   and ys :: "'c"   and y :: "'f"   and binp' :: "'j \<Rightarrow> ('a, 'b, 'c, 'f, 'e) qAbs set option"   and binp :: "'j \<Rightarrow> 'h option" shows "gFreshBinp MOD ys y binp' binp \<equiv> liftAll2 (gFreshAbs MOD ys y) binp' binp"
length_delete_index: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "length (delete_index n xs) = length xs - 1"
wlog_line_x_axis: fixes H :: "circline"   and P :: "circline \<Rightarrow> bool" assumes "is_poincare_line H"   and "P x_axis"   and "\<And>M. \<lbrakk>unit_disc_fix M; P (moebius_circline M H)\<rbrakk> \<Longrightarrow> P H" shows "P H"
g_def: shows "g \<equiv> \<lambda>j. if j \<le> k - 1 then f j else if j = k then b else f (j - 1)"
Sup_iter_unfold: fixes x :: "'a" shows "x ^ 0 \<squnion> (\<Squnion>n. x ^ Suc n) = \<Squnion> range ((^) x)"
hr_comp_invalid: fixes R1 :: "'c \<Rightarrow> 'b \<Rightarrow> assn"   and R2 :: "('c \<times> 'a) set" shows "hr_comp (invalid_assn R1) R2 = invalid_assn (hr_comp R1 R2)"
pos_less_divide_eq: 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 c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (divide b c) = less (times a c) b"
monic_prod_gcd: fixes A :: "'b set"   and f :: "'b poly"   and h :: "'b poly" assumes "finite A"   and "f \<noteq> 0" shows "monic (\<Prod>c\<in>A. gcd f (h - [:c:]))"
mat_add_assoc: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "f \<oplus> g \<oplus> h = f \<oplus> (g \<oplus> h)"
elims: fixes x :: "'c tree"   and y :: "'c tree" assumes "mirror x = y" obtains "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | l :: "'c tree" and xa :: "'c" and r :: "'c tree" where "x = \<langle>l, xa, r\<rangle>" and "y = \<langle>mirror r, xa, mirror l\<rangle>"
F2set1_JF2set: fixes c :: "'d"   and a :: "'d JF2" assumes "c \<in> F2set1 (dtor2 a)" shows "c \<in> JF2set a"
map_pmf_def: fixes f :: "'f \<Rightarrow> 'e"   and M :: "'f pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
assms_2: shows "P * Q = 1\<^sub>m n"
IH: fixes x :: "complex" assumes "0 < Re x" shows "(deriv ^^ i) (stirling_sum j m) x = stirling_sum (i + j) m x"
binop: shows "binop \<int>\<^sub>\<circ> vint_plus"
simps: fixes a1 :: "('state, 'effect) L_state"   and a2 :: "('state, 'effect) L_state" shows "L_bisimilar a1 a2 = ((\<exists>P F Q. a1 = EF (F, P) \<and> a2 = EF (F, Q) \<and> P \<sim>\<cdot>[F] Q) \<or> (\<exists>P F Q f. a1 = AC (f, F, \<langle>f\<rangle>P) \<and> a2 = AC (f, F, \<langle>f\<rangle>Q) \<and> P \<sim>\<cdot>[F] Q \<and> f \<in>\<^sub>f\<^sub>s F))"
sep_empty_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_empty zero zero"
of_int_le_of_int_power_cancel_iff: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "int"   and w :: "nat"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times (ring_1.of_int one plus zero minus b) w) (ring_1.of_int one plus zero minus x) = (b ^ w \<le> x)"
UniqueSucState_Target_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto"   and T :: "'d"   and U :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "A \<in> SAs (HA ST)"   and "A \<in> the (CompFun (HA ST) S)"   and "T \<in> States A"   and "U \<in> States A"   and "T \<in> StepConf (HA ST) (Conf ST) TS"   and "T \<noteq> U" shows "U \<notin> Target TS"
tendsto_exp_limit_at_right: fixes x :: "real" shows "((\<lambda>y. (1 + x * y) powr (1 / y)) \<longlongrightarrow> exp x) (at_right 0)"
sum_nonneg_eq_0_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> {#} \<subseteq># f x" shows "(comm_monoid_add.sum (+) {#} f A = {#}) = (\<forall>x\<in>A. f x = {#})"
inj_on_fun_updI: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and y :: "'e"   and x :: "'d" assumes "inj_on f A"   and "y \<notin> f ` A" shows "inj_on (f(x := y)) A"
prod_fun_unfold_prod: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and k :: "'b" assumes "finite {a. f a \<noteq> (0::'c)}"   and "finite {b. g b \<noteq> (0::'c)}" shows "prod_fun f g k = (\<Sum>(a, b). f a * g b when k = a + b)"
COD_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "monoidal_language.Cod C (free_monoidal_category.rep C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g)) = monoidal_language.Cod C (free_monoidal_category.rep C f) \<^bold>\<otimes> monoidal_language.Cod C (free_monoidal_category.rep C g)"
times_enat_simps_3: fixes n :: "nat" shows "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
that: fixes c' :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p (subtopology X (S - U)) (T - U) c'"   and "homologous_rel p (subtopology X S) T c c'" shows "thesis"
diff_ge_0_iff_ge: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and uminus :: "'r \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and a :: "'r"   and b :: "'r" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
imply_append: fixes ps :: "('a, 'b) fm list"   and qs :: "('a, 'b) fm list"   and r :: "('a, 'b) fm" shows "(ps @ qs \<^bold>\<leadsto> r) = (ps \<^bold>\<leadsto> qs \<^bold>\<leadsto> r)"
sel_3: fixes x21 :: "'f"   and x22 :: "'f list" shows "tl (x21 # x22) = x22"
wpo_s_imp_ns: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term" assumes "s \<succ> t" shows "s \<succeq> t"
dominoes_tile_matrix: fixes m :: "nat"   and n :: "nat" shows "{0..<m} \<times> {0..<2 * n} \<in> tiling domino"
collapse: fixes pheap :: "pheap" shows "pHeap (heapOf pheap) (addrOf pheap) = pheap"
map_iso_def: fixes f :: "('a, 'b, 'f, 'e) digraph_isomorphism" shows "map_iso f \<equiv> \<lparr>edge_rev = wrap_iso_arcs f (edge_rev M), edge_succ = wrap_iso_arcs f (edge_succ M)\<rparr>"
cod_in_codomains: fixes f :: "('o, 'a) arr" assumes "codomains f \<noteq> {}" shows "cod f \<in> codomains f"
basg_def: shows "basg \<equiv> \<lambda>i. if i \<in> basf ` Basis then inv_into Basis basf i else (0::'a, 1)"
poly_fun_times: fixes f :: "int \<Rightarrow> int"   and g :: "int \<Rightarrow> int" assumes "poly_fun f"   and "poly_fun g" shows "poly_fun (\<lambda>x. f x * g x)"
inject: fixes a :: "'g"   and b :: "'h"   and a' :: "'g"   and b' :: "'h" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
that: fixes p :: "linear_poly"   and c :: "rat" assumes "dim_poly p \<le> d"   and "x = p + lp_monom c d" shows "thesis"
hseq_leg\<^sub>0: shows "hseq t\<^sub>0 \<pi>.chine"
entails_\<G>_L_def: fixes N1 :: "('f \<times> 'l) set"   and N2 :: "('f \<times> 'l) set" shows "(N1 \<Turnstile>\<inter>\<G>L N2) = (\<forall>q\<in>Q. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2)))"
simps_2: fixes g :: "'f \<times> 'd set"   and gs :: "('f \<times> 'd set) list"   and c :: "('d, 'e, 'f) com" shows "guards (g # gs) c = Guard (fst g) (snd g) (guards gs c)"
distinct_3: fixes x1 :: "'b"   and x3 :: "'b" shows "left x1 \<noteq> mid x3"
simps_1: fixes x :: "name" shows "lookup [] x = Var x"
preservation2: fixes y :: "'a"   and x :: "'a" assumes "y ; x \<le> y" shows "(x ; y)\<^sup>\<star> ; x \<le> x ; y\<^sup>\<star>"
sams_lea2__sams: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A' B' C' D' E' F'"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.LeA Bet Cong D E F D' E' F'" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"
unsigned_def: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero \<equiv> map_fun Word.rep id (semiring_1.of_nat one plus zero \<circ> nat \<circ> take_bit LENGTH('l))"
as_needed_asses_submap_exec_vii: fixes f :: "('c, 'd) fmap"   and g :: "('c, 'd) fmap" assumes "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v" shows "f \<subseteq>\<^sub>f g"
append_atomics_reusables_neq_Nil: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "fst w \<le> snd w"   and "local.valid t"   and "l t \<le> fst w"   and "r t \<le> snd w" shows "local.atomics (max (fst w) (Suc (r t))) (snd w) @ reusables t w \<noteq> []"
\<Delta>\<Phi>_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" shows "\<Phi> (merge h1 h2) - \<Phi> h1 - \<Phi> h2 \<le> log 2 (real (size_hp h1 + size_hp h2 + 1)) + 1"
\<R>_def: shows "\<R> \<equiv> {Regions.region X I r |I r. Regions.valid_region X k I r}"
tuple_decode_encode': fixes xs :: "nat list"   and n :: "nat" assumes "length xs = Suc n" shows "tuple_decode n (tuple_encode n xs) = xs"
ParComm: fixes P :: "ccs"   and Q :: "ccs" shows "P \<parallel> Q \<equiv>\<^sub>s Q \<parallel> P"
nrename_ImpL': fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and M :: "trm"   and w :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = ImpL <c>.M w.N u"   and "c \<sharp> (R, N)"   and "w \<sharp> (R, M, u, x, y)"   and "x \<noteq> y" shows "(\<exists>M' N'. R = ImpL <c>.M' w.N' u \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N) \<or> (\<exists>M' N'. R = ImpL <c>.M' w.N' x \<and> y = u \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N)"
zero_less_power_abs_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times (abs a) n) = (a \<noteq> zero \<or> n = 0)"
not_dbm_le_lt_impl: fixes a :: "'d"   and b :: "'d" assumes "\<not> Le a \<prec> Lt b" shows "b \<le> a"
decompose_None: fixes f :: "'c"   and ss :: "('c, 'd) Term.term list"   and g :: "'c"   and ts :: "('c, 'e) Term.term list" assumes "decompose (Fun f ss) (Fun g ts) = None" shows "f \<noteq> g \<or> length ss \<noteq> length ts"
two_cases: fixes c :: "one\<^sub>\<bottom>" obtains "c = many"   | "c = once"   | "c = none"
is_generalisation_of_i_lt_arity: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "i < Arity t"
rel_transfer: fixes Sa :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sc :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((Sa ===> Sc ===> (=)) ===> list_all2 Sa ===> list_all2 Sc ===> (=)) list_all2 list_all2"
simps_3: fixes ao :: "'a option"   and x :: "'a"   and xs :: "'a list" shows "Compl_set_less_eq_aux ao (x # xs) [] = (\<not> proper_interval ao (Some x) \<and> Compl_set_less_eq_aux (Some x) xs [])"
cr_pmf_def: shows "pmf_as_function.cr_pmf \<equiv> \<lambda>x y. x = pmf y"
comp_by_index_inj1: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "comp_by_index x1 y1 = comp_by_index x2 y2" shows "x1 = x2"
a_repr_independence: fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H +> x"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "H +> x = H +> y"
blinding_of_on_axioms: shows "blinding_of_on UNIV rh bo"
noE: shows "Ifm bs (qe (list_conj (snd (partition bound0 (conjuncts p))))) = Ifm bs (E (list_conj (snd (partition bound0 (conjuncts p)))))"
possible_steps_alt2: 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 d :: "nat"   and t :: "transition" shows "(possible_steps e s r l i = {|(d, t)|}) = (ffilter (\<lambda>((origin, dest), t). Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)) (ffilter (\<lambda>((origin, dest), t). origin = s) e) = {|((s, d), t)|})"
LeftDerivationIntro_propagate: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro (\<alpha> @ \<delta>) i r ix D j \<gamma>"   and "i < length \<alpha>"   and "is_nonterminal (\<gamma> ! j)" shows "\<exists>\<omega>. LeftDerivation \<alpha> ((i, r) # D) \<omega> \<and> \<gamma> = \<omega> @ \<delta> \<and> j < length \<omega>"
maximal_def: fixes S :: "'b set"   and C :: "'b set set" shows "maximal S C \<equiv> \<forall>S'\<in>C. S \<subseteq> S' \<longrightarrow> S = S'"
sub_set_sysI: fixes \<U> :: "'a set"   and \<A> :: "'a set multiset" assumes "\<U> \<subseteq> \<V>"   and "\<A> \<subseteq># \<B>" shows "sub_set_system \<U> \<A> \<V> \<B>"
cblinfun_image_INF_eq_general: fixes Uinv :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and V :: "'c \<Rightarrow> 'b ccsubspace" assumes "Uinv o\<^sub>C\<^sub>L U o\<^sub>C\<^sub>L Uinv = Uinv"   and "U o\<^sub>C\<^sub>L Uinv o\<^sub>C\<^sub>L U = U"   and "\<And>i. V i \<le> Uinv *\<^sub>S \<top>" shows "U *\<^sub>S \<Sqinter> (range V) = (INF i. U *\<^sub>S V i)"
assms_2: shows "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1"
1: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace>) in v]"
K_iso: shows "\<forall>i\<in>I. K i \<in> Group.iso ((f \<circ> Gs) i) (Gs i)"
exhaust: fixes y :: "signs" obtains "y = Pos"   | "y = Neg"
items_eq_def: fixes k :: "nat"   and I :: "item set" shows "items_eq k I = {x \<in> I. item_end x = k}"
plus_subset1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "\<Omega>[F](g) \<subseteq> \<Omega>[F](\<lambda>x. f x + g x)"
sin_add: fixes x :: "'a"   and y :: "'a" shows "sin (x + y) = sin x * cos y + cos x * sin y"
mult_oner: fixes x :: "'b \<times> 'c" shows "times_prod' x one_prod = x"
has_derivative_compose: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and g :: "'d \<Rightarrow> 'e"   and g' :: "'d \<Rightarrow> 'e" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x))" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
p1: fixes l :: "nat" shows "tensor_P (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) (1\<^sub>m K) * tensor_P (proj_psi_l (l + 1)) (proj_k l) * tensor_P (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) (1\<^sub>m K) = tensor_P (proj_psi'_l l) (proj_k l)"
shift1_domI: fixes args :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv args" shows "shift1_dom args"
owner_seskey_shakey: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey"   and "(Owner n, SesKey SK) \<in> s" shows "SesKey SK \<notin> spied s"
cdf_diff_eq: fixes x :: "real"   and y :: "real" assumes "x < y" shows "cdf M y - cdf M x = Sigma_Algebra.measure M {x<..y}"
hom_one: shows "coeff 1 0 = (1::'b)"
iso_iff_mono_and_retraction: fixes f :: "'a" shows "B.iso f = (B.mono f \<and> B.retraction f)"
no_state_change_if_only_nonregular_events: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a"   and st :: "'b" assumes "trace c t c'"   and "\<nexists>ev. ev \<in> set t \<and> regular_event ev \<and> occurs_on ev = p"   and "states c p = st" shows "states c' p = st"
node_ptr_no_document_ptr_cast: fixes document_ptr :: "(_) document_ptr" shows "\<not> is_node_ptr_kind (cast document_ptr)"
Hom_Op: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" shows "Hom\<^bsub>Op C\<^esub> X Y = Hom\<^bsub>C\<^esub> Y X"
bdd_lookup_bddinsert: fixes w :: "bool list"   and bd :: "'a bdd"   and v :: "bool list"   and y :: "'a" assumes "bddh (length w) bd"   and "length w = length v" shows "bdd_lookup (bddinsert bd w y) v = (if w = v then y else bdd_lookup bd v)"