hcju/isabelleps huggingface dataset:
assms: shows "\<turnstile> (\<^bold>\<not> conjoin G (\<^bold>\<not> p))"
path_image_part_circlepath_same_Re: fixes b :: "real"   and a :: "real"   and r :: "real"   and c :: "complex" assumes "0 \<le> b"   and "b \<le> pi"   and "a = - b"   and "0 \<le> r" shows "path_image (part_circlepath c r a b) = sphere c r \<inter> {s. Re c + r * cos a \<le> Re s}"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
absorb2: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" shows "a \<union># b = b"
Cuts_def: shows "Cuts = {z. \<exists>C Delta Gamma. z = (Gamma @ Delta, {C # Gamma, FNot C # Delta})}"
in_restrict_modesD: fixes i :: "nat"   and mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list"   and x :: "'Var\<^sub>C"   and X :: "'Var\<^sub>C set"   and m :: "Mode" assumes "i < length mdss"   and "x \<in> (conc.restrict_modes mdss X ! i) m" shows "x \<in> X \<and> x \<in> (mdss ! i) m"
subsets_sat_core_def: fixes s :: "('a, 'b) state" shows "subsets_sat_core s = (\<forall>I\<subset>set (the (\<U>\<^sub>c s)). \<exists>v. (I, v) \<Turnstile>\<^sub>i\<^sub>s\<^sub>e s)"
simps_4: fixes X :: "tyvrs"   and T1 :: "ty"   and T2 :: "ty" assumes "X \<sharp> T1" shows "size_ty (\<forall>X<:T1. T2) = size_ty T1 + size_ty T2 + 1"
parallel_fixp_induct: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'd" assumes "class.ccpo luba orda (mk_less orda)"   and "class.ccpo lubb ordb (mk_less ordb)"   and "ccpo.admissible (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. P (fst x) (snd x))"   and "monotone orda orda f"   and "monotone ordb ordb g"   and "P (luba {}) (lubb {})"   and "\<And>x y. P x y \<Longrightarrow> P (f x) (g y)" shows "P (ccpo.fixp luba orda f) (ccpo.fixp lubb ordb g)"
that: fixes x :: "'a"   and y :: "'a" assumes "p = x * y"   and "x dvd a"   and "y dvd b" shows "thesis"
simps_2: fixes n :: "nat"   and ns :: "nat list"   and c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "c \<rightarrow>\<^bsub>n # ns\<^esub> c' = (\<exists>c''. c \<leadsto>\<^bsub>n\<^esub> c'' \<and> c'' \<rightarrow>\<^bsub>ns\<^esub> c')"
induct: fixes P :: "('b \<times> 'b) set \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "('b \<times> 'b) set"   and a1 :: "'b list" assumes "\<And>R xs. \<lbrakk>\<And>x xa. \<lbrakk>xs \<noteq> []; x \<in> set_pmf (pmf_of_set {..<length xs}); xa = xs ! x\<rbrakk> \<Longrightarrow> P R (lesss R xa xs); \<And>x xa. \<lbrakk>xs \<noteq> []; x \<in> set_pmf (pmf_of_set {..<length xs}); xa = xs ! x\<rbrakk> \<Longrightarrow> P R (greaters R xa xs)\<rbrakk> \<Longrightarrow> P R xs" shows "P a0 a1"
dependent_alt: fixes B :: "'b set" shows "dependent B = (\<exists>X. finite {x. X x \<noteq> (0::'a)} \<and> {x. X x \<noteq> (0::'a)} \<subseteq> B \<and> (\<Sum>x | X x \<noteq> (0::'a). X x *s x) = (0::'b) \<and> (\<exists>x. X x \<noteq> (0::'a)))"
det_four_block_mat_lower_left_zero: fixes A1 :: "'a mat"   and n :: "nat"   and A2 :: "'a mat"   and m :: "nat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat n m"   and "A3 = 0\<^sub>m m n"   and "A4 \<in> carrier_mat m m" shows "Determinant.det (four_block_mat A1 A2 A3 A4) = Determinant.det A1 * Determinant.det A4"
a_le: shows "a \<le> mod_type_class.from_nat (i - 1)"
r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #> x \<subseteq> carrier G"
matchC_C_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchC_C theta c d \<equiv> \<forall>s t c' s'. s \<approx> t \<and> (c, s) \<rightarrow>c (c', s') \<longrightarrow> (\<exists>d' t'. (d, t) \<rightarrow>c (d', t') \<and> s' \<approx> t' \<and> (c', d') \<in> theta)"
step_3_main_similar: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n"   and "0 < k" shows "similar_mat (step_3_main n k A) A"
UNION_positions_of: shows "(\<Union>x\<in>set (diag_mat A). {i. A $$ (i, i) = x \<and> i < length (diag_mat A)}) = {0..<length (diag_mat A)}"
rel: shows "SourceTerm SP \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm SQ"
is_adj_map_def: fixes ps :: "nat \<Rightarrow> nat list" shows "is_adj_map ps \<equiv> \<forall>u. distinct (ps u) \<and> set (ps u) = E `` {u} \<union> E\<inverse> `` {u}"
map\<^sub>0_in_hhom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<sigma>\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (G a)\<guillemotright>"
set_rotate: fixes n :: "nat"   and xs :: "'d list" shows "set (rotate n xs) = set xs"
self_append_conv: fixes xs :: "'e list"   and ys :: "'e list" shows "(xs = xs @ ys) = (ys = [])"
i_aggregate_i_take_mult: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "f_aggregate (f \<Down> (n * k)) k ag = i_aggregate f k ag \<Down> n"
borel_measurable_ereal_add: fixes f :: "'b \<Rightarrow> ereal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ereal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
distinct_3: fixes x1 :: "nat"   and x31 :: "'a convnet"   and x32 :: "'a convnet" shows "Input x1 \<noteq> Pool x31 x32"
add_block_fin_cond: fixes b :: "'a set" assumes "b \<subseteq> \<V>" shows "finite_incidence_system \<V> (add_block b)"
assms_3: shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S"
mset_le_distrib: fixes X :: "'c multiset"   and A :: "'c multiset"   and B :: "'c multiset" assumes "X \<subseteq># A + B" obtains Xa :: "'c multiset" and Xb :: "'c multiset" where "X = Xa + Xb" and "Xa \<subseteq># A" and "Xb \<subseteq># B"
mset_upt: fixes m :: "nat"   and n :: "nat" shows "mset [m..<n] = mset_set {m..<n}"
f: shows "\<forall>n c a. f c a $ n = (of_nat n + k c) * a $ n"
Composition: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<Psi> \<otimes> \<Psi>'' \<simeq> \<Psi>' \<otimes> \<Psi>''"
set_insort_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "set (linorder.insort_insert_key less_eq (\<lambda>x. x) x xs) = insert x (set xs)"
prob_space_axioms: fixes i :: "'a" shows "prob_space (if i \<in> I then M i else count_space {undefined})"
A: shows "(\<lambda>x. stock_measure (case x of 0 \<Rightarrow> t | Suc x \<Rightarrow> \<Gamma> x)) = case_nat (stock_measure t) (\<lambda>x. stock_measure (\<Gamma> x))"
LIM_Re: fixes f :: "'a \<Rightarrow> complex"   and L :: "complex"   and a :: "'a" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>x. Re (f x)) \<midarrow>a\<rightarrow> Re L"
subprob_measure_le_1: fixes M :: "'c measure"   and X :: "'c set" assumes "subprob_space M" shows "Sigma_Algebra.measure M X \<le> 1"
right_totalE: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" assumes "right_total A" obtains x :: "'a" where "A x y"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd" shows "Poly_Mapping.keys x = {k. poly_mapping.lookup x k \<noteq> (0::'d)}"
infsafety_def: fixes A :: "'b set"   and P :: "'b llist set" shows "infsafety A P \<equiv> \<forall>t\<in>A\<^sup>\<omega>. (\<forall>r\<in>finpref A t. \<exists>s\<in>A\<^sup>\<omega>. r @@ s \<in> P) \<longrightarrow> t \<in> P"
midpoint_eq_endpoint_2: fixes a :: "'b"   and b :: "'b" shows "(midpoint a b = b) = (a = b)"
eql_pos_lit_eql_atm: fixes A' :: "'a"   and \<eta> :: "'s"   and A :: "'a" shows "(Pos A' \<cdot>l \<eta> = Pos A) = (A' \<cdot>a \<eta> = A)"
flrestriction_frrestriction: fixes r :: "V"   and A :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<bullet> A) \<restriction>\<^sup>r\<^sub>\<bullet> A = r \<restriction>\<^sub>\<bullet> A"
assms_3: shows "\<Union> (Range a) \<subseteq> \<Omega>"
domain: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (pcr_alist T Ta) = (\<lambda>x. \<exists>y. list_all2 (rel_prod T Ta) x y \<and> (distinct \<circ> list.map fst) y)"
power_le_star: fixes x :: "'a"   and n :: "nat" shows "x\<^bsup>n\<^esup> \<le> x\<^sup>\<star>"
card_insert_if: fixes A :: "'e set"   and x :: "'e" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
cases: fixes a1 :: "'b \<Rightarrow> ('c, 'd) trm"   and a2 :: "('c + 'b, 'e, 'd) hp" assumes "NPadmit a1 a2" obtains \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and a :: "'d" where "a1 = \<sigma>'__" and "a2 = $\<alpha> a__"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and a :: "('c + 'b, 'e, 'd) hp" and b :: "('c + 'b, 'e, 'd) hp" where "a1 = \<sigma>''__" and "a2 = (aa__ ;; b__)" and "NPadmit \<sigma>''__ aa__" and "NPadmit \<sigma>''__ b__" and "PUadmitFO \<sigma>''__ b__ (BVP (PsubstFO aa__ \<sigma>''__))" and "hpsafe (PsubstFO aa__ \<sigma>''__)"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and a :: "('c + 'b, 'e, 'd) hp" where "a1 = \<sigma>'''__" and "a2 = ab__**" and "NPadmit \<sigma>'''__ ab__" and "PUadmitFO \<sigma>'''__ ab__ (BVP (PsubstFO ab__ \<sigma>'''__))" and "hpsafe (PsubstFO ab__ \<sigma>'''__)"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and ODE :: "('c + 'b, 'd) ODE" and \<phi> :: "('c + 'b, 'e, 'd) formula" where "a1 = \<sigma>''''__" and "a2 = EvolveODE ODE \<phi>'__" and "OadmitFO \<sigma>''''__ ODE (BVO ODE)" and "NFadmit \<sigma>''''__ \<phi>'__" and "FUadmitFO \<sigma>''''__ \<phi>'__ (BVO ODE)" and "fsafe (FsubstFO \<phi>'__ \<sigma>''''__)" and "osafe (OsubstFO ODE \<sigma>''''__)"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and a :: "('c + 'b, 'e, 'd) hp" and b :: "('c + 'b, 'e, 'd) hp" where "a1 = \<sigma>'''''__" and "a2 = (a \<union>\<union> b)" and "NPadmit \<sigma>'''''__ a" and "NPadmit \<sigma>'''''__ b"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<theta> :: "('c + 'b, 'd) trm" and x :: "'d" where "a1 = \<sigma>''''''__" and "a2 = (x__ := \<theta>__)" and "TadmitFO \<sigma>''''''__ \<theta>__"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<theta> :: "('c + 'b, 'd) trm" and x :: "'d" where "a1 = \<sigma>'''''''__" and "a2 = DiffAssign x \<theta>" and "TadmitFO \<sigma>'''''''__ \<theta>"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<phi> :: "('c + 'b, 'e, 'd) formula" where "a1 = \<sigma>" and "a2 = ? \<phi>" and "NFadmit \<sigma> \<phi>"
not_simple\<^sub>s\<^sub>t\<^sub>p_elim: fixes x :: "('c, 'd) strand_step" assumes "\<not> simple\<^sub>s\<^sub>t\<^sub>p x" shows "(\<exists>f T. x = send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t) \<or> (\<exists>a t t'. x = \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t) \<or> (\<exists>X F. x = \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<and> (\<nexists>\<I>. ineq_model \<I> X F))"
ActiveProcessSilentDecisionValuesIncrease: fixes sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and p :: "'a"   and q :: "'a"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and c :: "('a, 'c, 'b) configuration"   and msg :: "('a, 'c) message"   and c' :: "('a, 'c, 'b) configuration"   and start :: "'a \<Rightarrow> 'b" assumes "flpSystem sends"   and "p = q"   and "asynchronousSystem.steps trans sends c msg c'"   and "isReceiverOf p msg"   and "asynchronousSystem.initReachable trans sends start c" shows "{v. flpSystem.pSilDecVal trans sends start v q c} \<subseteq> {v. flpSystem.pSilDecVal trans sends start v q c'}"
that: fixes T :: "'a set" assumes "T \<in> B"   and "S \<subseteq> T"   and "S \<inter> \<Union> (B - {T}) = {}" shows "thesis"
reduced_row_echelon_form_upt_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "reduced_row_echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)) \<and> (\<forall>i. i < i + (1::'d) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'d)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'d)) $ n \<noteq> (0::'b))) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b))))"
borel_measurable_mono: fixes f :: "real \<Rightarrow> real" assumes "mono f" shows "f \<in> borel_measurable borel"
angle_action_defs: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and w :: "nat \<Rightarrow> 'b" shows "(\<diamond>\<langle>P\<rangle>_v) w = (\<exists>i. P (w |\<^sub>s i) \<and> ((w |\<^sub>s i) \<Turnstile> v$ \<noteq> $v))"
cr_transaction\<^sub>h_def: shows "transaction_merkle.cr_transaction\<^sub>h \<equiv> \<lambda>x y. x = the_Transaction\<^sub>h y"
mult_assoc: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" shows "a \<otimes> b \<otimes> c = a \<otimes> (b \<otimes> c)"
assoc_in_hom: shows "\<guillemotleft>\<a>[f, a, a] : (f \<star> a) \<star> a \<Rightarrow>\<^sub>S f \<star> a \<star> a\<guillemotright>"
max: fixes a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "a \<in> G"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" shows "rword (spp_of a) (spp_of g')"
ffb_iffb_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'c set" assumes "Inf_pres \<phi>" shows "(fb\<^sup>-\<^sub>\<F> \<phi> = f) = (fb\<^sub>\<F> f = \<phi>)"
total: fixes x :: "'a"   and y :: "'a" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
fmpred_cases: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and x :: "'a" assumes "fmpred P m" obtains "fmlookup m x = None"   | y :: "'b" where "fmlookup m x = Some y" and "P x y"
index_update_vec1: fixes i :: "nat"   and v :: "'c vec"   and a :: "'c" assumes "i < dim_vec v" shows "(v |\<^sub>v i \<mapsto> a) $ i = a"
in_carrier: shows "[\<one>, \<ominus> x] \<in> carrier (poly_ring R)"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 cd-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' n'. \<lbrakk>n cd-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' \<longrightarrow>\<^bsub>cd\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'" shows "P x1 x2 x3"
no_position_replicate_num_args: fixes t :: "('a, 'b) tm"   and d :: "dir" shows "\<not> position_of t (replicate (num_args t) dir.Left @ [d])"
algebraic_imp_represents_irreducible: fixes x :: "'a" assumes "algebraic x" shows "\<exists>p. p represents x \<and> irreducible p"
right_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<sigma> :: "'b" assumes "weak_composition V H" shows "weak_composition.right V H \<sigma> \<equiv> \<lambda>\<mu>. H \<mu> \<sigma> \<noteq> partial_magma.null V"
map_poly_hom_coeff_lift: fixes a :: "'a" shows "map_poly hom [:a:] = [:hom a:]"
phi_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "x \<in> I"   and "y \<in> I" shows "phi x \<le> phi y"
do_load_action_def: fixes ract :: "('a, 'd) mem_load_action" shows "do_load_action ract = (\<lambda>s. case ract of mr_Ref r f \<Rightarrow> mv_Ref (Option.bind (heap s r) (\<lambda>obj. obj_fields obj f)) | mr_Payload r f \<Rightarrow> mv_Payload (Option.bind (heap s r) (\<lambda>obj. obj_payload obj f)) | mr_Mark r \<Rightarrow> mv_Mark (map_option obj_mark (heap s r)) | mr_Phase \<Rightarrow> mv_Phase (phase s) | mr_fM \<Rightarrow> mv_Mark (Some (fM s)) | mr_fA \<Rightarrow> mv_Mark (Some (fA s)))"
strict_mono_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "order.strict_mono less f" shows "order.mono less_eq f"
count_sum_mset: fixes M :: "'b multiset multiset"   and b :: "'b" shows "count (\<Sum>\<^sub># M) b = (\<Sum>P\<in>#M. count P b)"
unit_factor_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "(unit_factor a = zero) = (a = zero)"
V_not_in_TOTAL: shows "V_constotal \<notin> TOTAL"
residual_flow_attainability: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "wf_residual_network"   and "flow \<Delta> f" shows "flow_attainability (residual_network f)"
from_hma\<^sub>m_def: fixes a :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a = Matrix.mat CARD('d) CARD('c) (\<lambda>(i, j). a $h mod_type_class.from_nat i $h mod_type_class.from_nat j)"
Ostrowski_elem_Suc: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a"   and j :: "nat" assumes "vals_nonequiv K n vv"   and "x \<in> carrier K"   and "Ostrowski_elem K n vv x"   and "j \<in> nset (Suc 0) n" shows "0 < vv j x"
dom_cod: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "A1xA2.dom (A1xA2.cod f) = A1xA2.cod f"
TR_isom_gch_units: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n f"   and "i \<le> n"   and "j \<le> n"   and "i < j" shows "card {k. k \<le> n \<and> f k \<cong> E} = card {k. k \<le> n \<and> cmp f (transpos i j) k \<cong> E}"
poly_roots_smult: fixes c :: "'b"   and p :: "'b poly" assumes "c \<noteq> (0::'b)" shows "poly_roots (Polynomial.smult c p) = poly_roots p"
is_ground_subst_def: fixes \<sigma> :: "'s" shows "is_ground_subst \<sigma> = (\<forall>A. is_ground_atm (A \<cdot>a \<sigma>))"
fmlookup_map: fixes f :: "'e \<Rightarrow> 'c"   and m :: "('d, 'e) fmap"   and x :: "'d" shows "fmmap f m $$ x = map_option f (m $$ x)"
cases: fixes a :: "'a itrm" assumes "CFp a" obtains x :: "dB" where "a = Pure xa__"   | f :: "'a itrm" and x :: "'a" where "a = f \<diamondop> Opaque x" and "CFp f"
idhom_left: fixes H :: "'v set" assumes "ImG \<subseteq> H" shows "id \<down> H \<circ> T = T"
elims_3: fixes x :: "nat"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "\<not> wadjust_goon_left_moving x xa xb" obtains m :: "nat" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = rs" and "xb = (l, r)" and "\<not> (wadjust_goon_left_moving_B m rs (l, r) \<or> wadjust_goon_left_moving_O m rs (l, r))"
lookup_default_filter: fixes d :: "'a"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('b, 'a) mapping"   and k :: "'b" shows "Mapping.lookup_default d (Mapping.filter P m) k = (if P k (Mapping.lookup_default d m k) then Mapping.lookup_default d m k else d)"
has_integral_nonneg: fixes f :: "'b \<Rightarrow> real"   and i :: "real"   and S :: "'b set" assumes "(f has_integral i) S"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "0 \<le> i"
cong_refl: fixes x :: "('a \<times> real) msllist"   and y :: "('a \<times> real) msllist"   and R :: "('a \<times> real) msllist \<Rightarrow> ('a \<times> real) msllist \<Rightarrow> bool" assumes "x = y" shows "msllist.v4.congclp R x y"
elims: fixes x :: "('b, 'a) Term.term"   and xa :: "('b, 'a) Term.term"   and y :: "('a \<Rightarrow> ('b, 'a) Term.term) option" assumes "mgu x xa = y" obtains s :: "('b, 'a) Term.term" and t :: "('b, 'a) Term.term" where "x = s" and "xa = t" and "y = (case unify [(s, t)] [] of None \<Rightarrow> None | Some res \<Rightarrow> Some (subst_of res))"
x2: fixes x :: "'a"   and xa :: "'a"   and xb :: "'a"   and xc :: "'a" assumes "t \<notin> set ts"   and "disjoint (insert t (set ts))"   and "x \<in> t"   and "xa \<in> t"   and "xa \<notin> s"   and "xb \<in> s"   and "xc \<in> s" shows "\<not> t - s \<subseteq> \<Union> (set (partList4 (s - t) ts))"
congruenceSubsetWeakBisim: shows "congruence \<subseteq> weakBisim"
inf2: fixes B :: "'d"   and k :: "'d"   and b :: "'d"   and a :: "'d" assumes "B \<equiv> inf k b" shows "inf a B \<equiv> inf k (inf a b)"
nonneg: shows "nonneg_cexpr (shift_var_set (shift_var_set V')) (case_nat t' (case_nat t \<Gamma>)) f"
mndet_is_process: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}" shows "is_process (\<Union>x\<in>A. F (x \<rightarrow> P x), \<Union>x\<in>A. D (x \<rightarrow> P x))"
less_mod_eq_imp_add_divisor_le: fixes x :: "nat"   and y :: "nat"   and m :: "nat" assumes "x < y"   and "x mod m = y mod m" shows "x + m \<le> y"
max_torder_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "max_torder r a b = (if Domainp r a \<and> Domainp r b then if r a b then b else a else if a = b then a else SOME a. \<not> Domainp r a)"
inject_2: fixes b :: "'b"   and b' :: "'b" shows "(Inr b = Inr b') = (b = b')"
prems_2: shows "\<nexists>n Ta. Fun (Val n) Ta \<sqsubseteq> Fun f T"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "partial_order L"   and "lower_semilattice_axioms L" shows "lower_semilattice L"
ord_pow_gen: fixes x :: "'a"   and k :: "nat" assumes "x \<in> carrier G" shows "ord (x [^] k) = (if k = 0 then 1 else ord x div gcd (ord x) k)"
coneToFun_mapsto: shows "coneToFun \<in> cones S.unity \<rightarrow> Pi I (S.set \<circ> D) \<inter> S.extensional' I"
FrQ: shows "extractFrame Q = \<langle>A\<^sub>Q', \<Psi>\<^sub>Q\<rangle>"
Rice_lm_2: fixes A :: "nat set"   and n :: "nat" assumes "index_set A"   and "A \<noteq> {}"   and "A \<noteq> UNIV"   and "n \<in> A"   and "nat_to_ce_set n = {}" shows "one_reducible_to univ_ce (- A)"
simps_8: fixes t :: "'c local.term" shows "Cod \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Cod t \<^bold>\<otimes> \<^bold>\<I>"
of_circline_def: shows "of_circline \<equiv> map_fun rep_circline abs_ocircline of_circline_clmat"
ListD: fixes x :: "('b, unit) Old_Datatype.node set" assumes "x \<in> List" shows "x \<in> list (range Leaf)"
sorted_filter: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "sorted (map f xs)" shows "sorted (map f (filter P xs))"
prj_in_hom_2: shows "\<guillemotleft>\<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>1 : HH\<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<nu>.apex\<guillemotright>"
AndE2: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form" assumes "G \<turnstile> And a b" shows "G \<turnstile> b"
prod_list_nonneg: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (0::'a) \<le> x" shows "(0::'a) \<le> prod_list xs"
eucl_of_list_inj: fixes xs :: "real list"   and ys :: "real list" assumes "length xs = DIM('a)"   and "length ys = DIM('a)"   and "eucl_of_list xs = eucl_of_list ys" shows "xs = ys"
subdomainI': fixes H :: "'a set" assumes "subring H R" shows "subdomain H R"
possworld_nec: fixes v :: "i"   and x :: "\<nu>" shows "[PossibleWorld (x\<^sup>P) \<^bold>\<equiv> \<^bold>\<box>PossibleWorld (x\<^sup>P) in v]"
self_financing_def: fixes Mkt :: "('c, 'd) discrete_market"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "self_financing Mkt p = (\<forall>n. val_process Mkt p (Suc n) = cls_val_process Mkt p (Suc n))"