hcju/isabelleps huggingface dataset:
set_disconnected_nodes_get_disconnected_nodes_is_l_set_disconnected_nodes_get_disconnected_nodes: shows "l_set_disconnected_nodes_get_disconnected_nodes DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c n p. (\<And>x. n \<le> x \<Longrightarrow> P p) \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (C v)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
complementary_def: fixes H1 :: "'a set"   and H2 :: "'a set" shows "complementary H1 H2 = (H1 \<inter> H2 = {\<one>})"
map_sel_2: fixes a :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "\<not> lnull a" shows "ltl (lmap f a) = lmap f (ltl a)"
that: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I"   and "a \<in>\<^sub>\<circ> f i" shows "thesis"
Abs_cfun_inverse2: fixes f :: "'c \<Rightarrow> 'd" assumes "cont f" shows "Rep_cfun (Abs_cfun f) = f"
distinct_1: fixes x1 :: "Keys"   and x2 :: "Secrets" shows "kKS x1 \<noteq> sKS x2"
assms_4: shows "h \<turnstile> append_child ptr' child \<rightarrow>\<^sub>h h'"
finite_not_eq_zero_sumI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {a. f a \<noteq> (0::'c)}"   and "finite {b. g b \<noteq> (0::'c)}" shows "finite {a + b |a b. f a \<noteq> (0::'c) \<and> g b \<noteq> (0::'c)}"
that: fixes e :: "real" assumes "0 < e"   and "b - a = e *\<^sub>R (x2 - x1)" shows "thesis"
lformula_NFOR: fixes \<Phi> :: "('a, 'k) aformula set" assumes "finite \<Phi>" shows "lformula (NFOR \<Phi>) = Ball \<Phi> lformula"
existential: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>"   and \<alpha> :: "'b" assumes "[\<phi> \<alpha> in v]" shows "[\<^bold>\<exists>\<alpha>. \<phi> \<alpha> in v]"
mod_ring_divide: shows "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (divide_p p) (/)"
position_augment: fixes n :: "nat"   and f :: "'a list \<Rightarrow> 'a"   and p :: "'a list" assumes "position ((augment_list f ^^ n) p)" shows "position p"
step_load_ubx_hit: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and \<tau> :: "Unboxed.type"   and x :: "'var"   and i :: "('dyn, 'ubx1, 'ubx2) unboxed"   and i' :: "'dyn"   and H :: "'henv"   and d :: "'dyn"   and blob :: "('dyn, 'ubx1, 'ubx2) unboxed"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = ILoadUbx \<tau> x"   and "cast_Dyn i = Some i'"   and "heap_get H (x, i') = Some d"   and "Subx.unbox \<tau> d = Some blob" shows "Subx.step (State F H (Frame f pc (i # \<Sigma>) # st)) (State F H (Frame f (Suc pc) (blob # \<Sigma>) # st))"
that: fixes p :: "'a list" assumes "mcp z W p" shows "thesis"
lb: fixes x :: "float" assumes "0 < x" shows "real_of_float (lb_sqrt prec x) \<le> sqrt (real_of_float x)"
card_le_dim_spanning: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "V \<subseteq> span B"   and "finite B"   and "card B \<le> dim V" shows "independent B"
empty_one_box_correct: fixes n :: "nat"   and x :: "nat"   and lm :: "nat list" shows "{\<lambda>nl. nl = 0 \<up> n @ x # lm} [Dec n 2, Goto 0] {\<lambda>nl. nl = 0 # 0 \<up> n @ lm}"
sign_r_pos_0: fixes x :: "real" shows "\<not> sign_r_pos 0 x"
NSBseq_def: fixes X :: "nat \<Rightarrow> 'a" shows "NSBseq X = (\<forall>N\<in>HNatInfinite. (*f* X) N \<in> HFinite)"
sigma_finite_measure_axioms: fixes xy :: "real \<times> 'a" shows "sigma_finite_measure (K.lim_stream xy)"
list_less_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs < ys) = ((xs, ys) \<in> lexord {(u, v). u < v})"
unbox_box: fixes x :: "'a" shows "unbox\<cdot>(box\<cdot>x) = up\<cdot>x"
less_trans: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<prec> y"   and "y \<prec> z" shows "x \<prec> z"
continuous_map_subtopology_eu: fixes S :: "'c set"   and T :: "'d set"   and h :: "'c \<Rightarrow> 'd" shows "continuous_map (top_of_set S) (top_of_set T) h = (continuous_on S h \<and> h ` S \<subseteq> T)"
max_p_div_calM: shows "\<not> p ^ Suc (multiplicity p m) dvd card calM"
K3_imp_Gets_evs: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and T2 :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Says A Tgs \<lbrace>Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "Gets A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"
simps: fixes rr :: "recf" shows "rec_maxr rr = (let vl = arity rr; rt = recf.id (Suc vl) (vl - 1); rf1 = Cn (Suc (Suc vl)) rec_le [recf.id (Suc (Suc vl)) (Suc vl), recf.id (Suc (Suc vl)) vl]; rf2 = Cn (Suc (Suc vl)) rec_not [Cn (Suc (Suc vl)) rr (get_fstn_args (Suc (Suc vl)) (vl - 1) @ [recf.id (Suc (Suc vl)) (Suc vl)])]; rf = Cn (Suc (Suc vl)) rec_disj [rf1, rf2]; Qf = Cn (Suc vl) rec_not [rec_all rt rf] in Cn vl (rec_sigma Qf) (get_fstn_args vl vl @ [recf.id vl (vl - 1)]))"
isometry_linear: fixes f :: "'a \<Rightarrow> 'a" assumes "f (0::'a) = (0::'a)"   and "\<forall>x y. dist (f x) (f y) = dist x y" shows "linear f"
lookup_not_edom: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b" assumes "x \<notin> edom m" shows "m x = \<bottom>"
rat_list_outside_zero: fixes cs :: "rat list"   and i :: "nat" assumes "length cs \<le> i" shows "Abstract_Linear_Poly.coeff (list_to_lpoly cs) i = 0"
cnf2bhv_lActive: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>\<langle>c \<and> t\<rangle> = the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1"
\<alpha>_def: shows "\<alpha> \<equiv> \<Squnion> (value_flow \<Delta> ` {g. flow \<Delta> g})"
basic_identity\<^sub>E_def: shows "basic_identity\<^sub>E \<equiv> \<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>))"
id_bnf_o: fixes f :: "'a \<Rightarrow> 'b" shows "BNF_Composition.id_bnf \<circ> f = f"
B: shows "B = one_dim_iso B *\<^sub>C id_cblinfun"
remove: fixes A :: "'e set"   and x :: "'e" assumes "finite A"   and "x \<in> A" shows "card A = Suc (card (A - {x}))"
P'B'Q': shows "(P', B', Q') = diagonal_to_Smith_i_PQ xs i bezout (P, B, Q)"
relpowp_E: fixes n :: "nat"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c" assumes "(P ^^ n) x z" obtains "n = 0" and "x = z"   | y :: "'c" and m :: "nat" where "n = Suc m" and "(P ^^ m) x y" and "P y z"
dom_comp: fixes g :: "'b"   and f :: "'b" assumes "J.seq g f" shows "J.dom (J.comp g f) = J.dom f"
P2L_L2P: fixes l :: "proj2_line" shows "P2L (L2P l) = l"
right_unit: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M" shows "composition a unit = a"
bounded_clinear_sub: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_clinear (\<lambda>x. f x - g x)"
section_is_mono: fixes g :: "'a" assumes "section g" shows "local.mono g"
fmlookup_dom'_iff: fixes x :: "'e"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "(x \<in> fmdom' m) = (\<exists>a. fmlookup m x = Some a)"
emeasure_space: fixes M :: "'b measure"   and A :: "'b set" shows "emeasure M A \<le> emeasure M (space M)"
segment_open_subset_closed: fixes a :: "'b"   and b :: "'b" shows "{a<--<b} \<subseteq> {a--b}"
eq_intr_ax_def: shows "eq_intr_ax \<equiv> (Core.A \<longmapsto> Core.B) \<longmapsto> (Core.B \<longmapsto> Core.A) \<longmapsto> mk_eq Core.A Core.B"
closed_GSgame_axioms_def: fixes A :: "'b llist set"   and N :: "enat" shows "GaleStewartDefensiveStrategies.closed_GSgame_axioms A N \<equiv> \<forall>e. e \<in> A \<longrightarrow> (\<exists>p. lprefix (llist_of p) e \<and> (\<forall>e'. lprefix (llist_of p) e' \<longrightarrow> llength e' = 2 * N \<longrightarrow> e' \<in> A))"
basic_monos_7: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
fps_is_unit_iff: fixes f :: "'a fps" shows "is_unit f = (fps_nth f 0 \<noteq> (0::'a))"
no_vs_change_valid_in_snapshot: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap"   and vs :: "'a set" assumes "as \<in> valid_plans PROB"   and "sat_precond_as s as"   and "vars_change as vs s = []" shows "as \<in> valid_plans (snapshot PROB (fmrestrict_set vs s))"
keys_eq_empty: fixes c :: "'a \<Rightarrow>\<^sub>0 'b" shows "(keys c = {}) = (c = 0)"
hull_insert: fixes P :: "'a set \<Rightarrow> bool"   and a :: "'a"   and S :: "'a set" shows "P hull insert a S = P hull insert a (P hull S)"
maxl_fun_gt: fixes x :: "'a" assumes "x \<in> carrier D"   and "\<exists>y\<in>carrier D. x \<preceq> y \<and> x \<noteq> y" shows "x \<preceq> maxl_fun D x \<and> maxl_fun D x \<noteq> x"
sub3_preserves_dimensions_2: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub3 sel_fun mf d l (r, A) = (r', A')" shows "dim_col A' = dim_col A"
d_implies_ts_process_traces: fixes P :: "'a process" assumes "deterministic P" shows "ts_process (traces P) = P"
eq: shows "([^]\<^bsub>homology_group 0 X\<^esub>) (homologous_rel_set 0 X {} (frag_of f)) = (\<lambda>n. homologous_rel_set 0 X {} (frag_cmul n (frag_of f)))"
io_path: shows "length ioV = length ioV' \<and> length ioV = length ioP \<and> path M2 ((ioV || ioV') || ioP) (initial M2) \<and> target ((ioV || ioV') || ioP) (initial M2) = q2"
not_primitiveE: fixes f :: "'a poly" assumes "\<not> primitive f" obtains x :: "'a" where "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y" and "\<not> x dvd (1::'a)"
14: shows "[a] @ b\<^sub>1 =\<^sub>F b\<^sub>1 @ [a]"
is_singletonI': fixes A :: "'d set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
sel_5: fixes x51 :: "nat"   and x52 :: "'d"   and x53 :: "'d" shows "occurs_on (RecvMarker x51 x52 x53) = x52"
distinct_path_subset: fixes G1 :: "('a, 'b) graph"   and G2 :: "('a, 'b) graph"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G1"   and "valid_unMultigraph G2"   and "edges G1 \<subseteq> edges G2"   and "nodes G1 \<subseteq> nodes G2"   and "valid_unMultigraph.is_trail G1 v ps v'" shows "valid_unMultigraph.is_trail G2 v ps v'"
Empty: fixes sp :: "'c set"   and A :: "'c set set" shows "{} \<in> sigma_sets sp A"
sorted_append_iff: fixes cmp :: "'b comparator"   and xs :: "'b list"   and ys :: "'b list" shows "Sorting_Algorithms.sorted cmp (xs @ ys) = (Sorting_Algorithms.sorted cmp xs \<and> Sorting_Algorithms.sorted cmp ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. compare cmp x y \<noteq> Greater))"
no_variables_aval: fixes a :: "vname aexp"   and s :: "vname \<Rightarrow> value option"   and s' :: "vname \<Rightarrow> value option" assumes "enumerate_aexp_inputs a = {}"   and "enumerate_regs a = {}" shows "aval a s = aval a s'"
G: shows "UNlift cl dl P F II1 = UNlift cl dl P F II2"
image_linear_greaterThan: fixes c :: "'c"   and b :: "'c"   and x :: "'c" assumes "c \<noteq> (0::'c)" shows "(\<lambda>x. c * x + b) ` {x<..} = (if (0::'c) < c then {c * x + b<..} else {..<c * x + b})"
one_le_inverse_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq one (inverse x) = (less zero x \<and> less_eq x one)"
ide_char: fixes a :: "'a" shows "\<tau>.A.ide a = (\<tau>.A.arr a \<and> \<tau>.A.dom a = a \<and> \<tau>.A.cod a = a)"
inducts: fixes x1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and x2 :: "'d"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and x4 :: "nat \<Rightarrow> nat"   and x5 :: "'e \<Rightarrow> nat"   and x6 :: "nat"   and x7 :: "'d"   and x8 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and P :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> ('e \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^bsub>x4,x5,x6\<^esub> \<langle>x7, x8\<rangle>"   and "\<And>A l D v n l' D' k. A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle> \<Longrightarrow> P A l D k v n l' (DBM_Normalization.norm (FW D' n) (\<lambda>x. real (k x)) n)" shows "P x1 x2 x3 x4 x5 x6 x7 x8"
steps_c\<^sub>1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Abrupt s) \<rightarrow>\<^sup>* (Skip, Abrupt s)"
fls_lr_inverse_base_1: fixes f :: "'a fls"   and x :: "'a" shows "fls_nth (fls_left_inverse f x) (- fls_subdegree f) = x"
one: fixes I :: "interp"   and \<theta> :: "trm"   and J :: "interp"   and \<eta> :: "trm" assumes "\<And>\<nu>. Uvariation \<nu> (F 0) {RVar x, DVar x} \<Longrightarrow> term_sem I \<theta> \<nu> = term_sem J \<eta> \<nu>"   and "solves_ODE I F x \<theta>" shows "solves_ODE J F x \<eta>"
IH: fixes keeps :: "('v \<times> 'v) list" shows "set (minimalize_offending_overapprox ff keeps G nP) \<subseteq> set ff \<union> set keeps"
antisym: fixes a :: "'a::order"   and b :: "'a::order" assumes "(a::'a::order) \<le> (b::'a::order)"   and "(b::'a::order) \<le> (a::'a::order)" shows "(a::'a::order) = (b::'a::order)"
IMP_PRE_def: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "IMP_PRE P1 P2 \<equiv> \<forall>x. P1 x \<longrightarrow> P2 x"
abs_Gcd_eq: fixes K :: "int set" shows "\<bar>Gcd K\<bar> = Gcd K"
ord_embed_def: fixes f :: "'c \<Rightarrow> 'd" shows "ord_embed f = (\<forall>x y. (f x \<le> f y) = (x \<le> y))"
iter_dom_properties2: fixes ss' :: "nat list list"   and w' :: "nat list" shows "iter f step start (unstables r step start) = (ss', w') \<longrightarrow> ss' \<in> list n A"
unfold2: fixes E :: "(\<AA> state \<times> \<AA> state \<Rightarrow> Set(\<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state"   and self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and POST' :: "(\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and BODY :: "(\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> Set(\<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self"   and "\<tau> \<Turnstile> true"   and "\<tau> \<Turnstile> POST' self"   and "\<And>res. (res \<triangleq> if self.boss@pre \<doteq> null then Set{self.salary@pre} else self.boss@pre.contents@pre()->including\<^sub>S\<^sub>e\<^sub>t(self.salary@pre) endif) = (POST' self and (res \<triangleq> BODY self))" shows "(\<tau> \<Turnstile> E (self.contents@pre())) = (\<tau> \<Turnstile> E (BODY self))"
subterm_of_ex_set_iff_subtermeqset: fixes t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" shows "(t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M) = (\<exists>t'\<in>M. subterm_of t t')"
mult_not_zero: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
alpha_preserves_qGood1: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "X #= X'"   and "qGood X" shows "qGood X'"
zero_rep: shows "\<zero> = (\<lambda>m. p_residue m 0)"
sup_relative_same_increasing: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<le> y"   and "x \<squnion> z = x \<squnion> w" shows "y \<squnion> z = y \<squnion> w"
bex_set_simps_1: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>set []. P x) = False"
a1: shows "LeftDerives1 (\<delta> @ \<alpha>) i r \<beta>"
gauss_jordan_transform: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and nc' :: "nat"   and A' :: "'c mat"   and B' :: "'c mat"   and b :: "'d" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc'"   and "gauss_jordan A B = (A', B')" shows "\<exists>P\<in>Units (ring_mat TYPE('c) nr b). A' = P * A \<and> B' = P * B"
restrict_prod_Suc: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and R :: "('f, 'g) Ring_scheme"   and J :: "nat \<Rightarrow> 'f set"   and B :: "nat \<Rightarrow> 'f set Ring"   and S :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and f :: "nat \<Rightarrow> 'f set" assumes "aGroup A"   and "\<forall>k\<le>Suc (Suc n). ideal R (J k)"   and "\<forall>k\<le>Suc (Suc n). B k = R /\<^sub>r J k"   and "\<forall>k\<le>Suc (Suc n). S k = pj R (J k)"   and "f \<in> carrier (r\<Pi>\<^bsub>{j. j \<le> Suc (Suc n)}\<^esub> B)" shows "restrict f {j. j \<le> Suc n} \<in> carrier (r\<Pi>\<^bsub>{j. j \<le> Suc n}\<^esub> B)"
tdev'_def: fixes p :: "nat"   and x :: "'b pdevs" shows "tdev' p x = sum_list' p (map (abs \<circ> snd) (list_of_pdevs x))"
CSP2_notok_b: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "P (a, b\<lparr>ok := False\<rparr>)" shows "CSP2 P (a, b\<lparr>ok := False\<rparr>)"
pre_arith_simps_3: fixes z :: "'h"   and x :: "'h"   and y :: "'h" shows "(z \<le> max x y) = (z \<le> x \<or> z \<le> y)"
insert_remove: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
FactGroup_hom: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "(\<lambda>X. the_elem (h ` X)) \<in> hom (G Mod kernel G H h) H"
D': fixes \<psi> :: "'a"   and b :: "'a" shows "norm (representation B' \<psi> b) \<le> norm \<psi> * D'"
arrE: fixes f :: "'a" assumes "hom'.arr f" shows "\<guillemotleft>f : trg (src f) \<rightarrow> trg e\<guillemotright>"
RSpan_single_closed: fixes m :: "'m" assumes "m \<in> M" shows "RSpan [m] \<subseteq> M"
R_def: shows "R \<equiv> (\<cdot>\<^sub>s\<^sub>e\<^sub>t) (subterms s \<union> subterms t - V)"
fold_Cons_rev: fixes xs :: "'g list" shows "fold (#) xs = (@) (rev xs)"
set_case_def: fixes s :: "'d set"   and c_empty :: "'c"   and c_sing :: "'d \<Rightarrow> 'c"   and c_else :: "'c" shows "set_case s c_empty c_sing c_else = (if s = {} then c_empty else if card s = 1 then c_sing (set_choose s) else c_else)"
NC: shows "\<not> cond xh \<longrightarrow> x = xh"
iIN_iprev: fixes n :: "nat"   and t :: "nat"   and d :: "nat" assumes "n < t"   and "t \<le> n + d" shows "iprev t [n\<dots>,d] = t - Suc 0"
a2c_def: shows "a2c = (\<Lambda> a. if a \<sqsubseteq> \<bottom> then none else many)"
elims_2: fixes x :: "('a, 'b, 'c) SLin_state \<times> ('a, 'b, 'c) SLin_state" assumes "P22 x" obtains s1 :: "('a, 'b, 'c) SLin_state" and s2 :: "('a, 'b, 'c) SLin_state" where "x = (s1, s2)" and "initialized s2 \<longrightarrow> dstate (f (s1, s2)) = dstate s2"
neg_sup_eq_inf: 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "uminus (sup a b) = inf (uminus a) (uminus b)"
p: shows "p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i"
assoc_props_5: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "trg \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<cdot>) prj0 prj1 f g h, Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr> = trg f"
lex_transI: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (lex r)"
UINF_as_Sup_collect': fixes f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> P \<bullet> f P) = \<Sqinter>range f"
stream_all_def: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" shows "stream_all P s = (\<forall>p. P (s !! p))"
SUP_bot_conv_2: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and B :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
doesnt_read_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var" shows "doesnt_read c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle> \<longrightarrow> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>) \<or> (\<forall>v. \<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>))"
vifintersection_vinsert_is: fixes I :: "V"   and j :: "V"   and f :: "V \<Rightarrow> V" assumes "I \<noteq> 0" shows "\<Inter>\<^sub>\<circ> (VLambda (vinsert j I) f `\<^sub>\<circ> vinsert j I) = f j \<inter>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
normal_form_correct: fixes w :: "nat \<Rightarrow> String.literal set"   and \<phi> :: "String.literal ltln" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n normal_form \<phi>"
simps_4: fixes f :: "('e, 'd) trm"   and g :: "('e, 'd) trm" shows "FVT (Plus f g) = FVT f \<union> FVT g"
funk_closure: fixes A :: "'f set" shows "{(x, x'). \<forall>a. x a = x' a} `` {x. range x \<subseteq> A} = {x. range x \<subseteq> A}"