hcju/isabelleps huggingface dataset:
inverse_ring_isomorphism: fixes \<eta> :: "'d \<Rightarrow> 'e"   and R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and R' :: "'e set"   and addition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multiplication' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero' :: "'e"   and unit' :: "'e" assumes "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "ring_isomorphism (restrict (inv_into R \<eta>) R') R' addition' multiplication' zero' unit' R addition multiplication zero unit"
part_equiv_def: fixes R :: "('a \<times> 'a) set" shows "part_equiv R = (sym R \<and> trans R)"
assms_3: shows "mod_type_class.to_nat a < k \<or> mod_type_class.to_nat b < k"
venv_approx_trans: fixes ve1 :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and ve2 :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and ve3 :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" assumes "ve1 \<lessapprox> ve2"   and "ve2 \<lessapprox> ve3" shows "ve1 \<lessapprox> ve3"
override_idem_implies_vwb: fixes X :: "'d \<Longrightarrow> 'e" assumes "mwb_lens X"   and "\<And>s. s \<triangleleft>\<^bsub>X\<^esub> s = s" shows "vwb_lens X"
irreducible_pos_gcd: fixes p :: "int poly"   and q :: "int poly" assumes "irreducible p"   and "0 < lead_coeff p" shows "gcd p q \<in> {1, p}"
splus_eq_splus: fixes t :: "('a, 'b) pp"   and u :: "('a, 'b) pp \<times> nat" shows "local.splus t u = nat_term_class.splus (t, the_min) u"
lookup_def: fixes al :: "('b, 'c) assoc_list" shows "Assoc_List.lookup al = map_of (assoc_list.impl_of al)"
Call_lemma: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and C' :: "char list"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M, b :  Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T \<and> P \<turnstile> C' \<preceq>\<^sup>* D' \<and> is_type P T' \<and> (\<forall>T\<in>set Ts'. is_type P T) \<and> wf_md P D' (M, b, Ts', T', m')"
word_set_bit_0: fixes n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit 0 n b = word_of_int (bin_sc n b 0)"
cases: fixes a :: "'b"   and s :: "'b + 'c" assumes "a \<in> Basic_BNFs.setl s" obtains x :: "'b" where "a = x" and "s = Inl x"
add_arc_in: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" assumes "wf_digraph G"   and "a \<in> arcs G" shows "pre_digraph.add_arc G a = G"
assert_spmf_simps_2: shows "assert_spmf False = return_pmf None"
rell_G_def: fixes L1 :: "'n \<Rightarrow> 'u \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'v \<Rightarrow> bool" shows "rell_G L1 L2 = rel_G L1 L2 (=) (=) (=) (=)"
Collect_congs_eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(Collect ((\<cong>) x) = Collect ((\<cong>) y)) = x \<cong> y"
charts_proj_space_def: shows "charts_proj_space = insert last_chart (basis_chart ` Basis)"
cases: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "('fun, 'var) Term.term" assumes "a1 \<turnstile>\<^sub>c a2" obtains t :: "('fun, 'var) Term.term" and M :: "('fun, 'var) Term.term set" where "a1 = Ma__" and "a2 = t" and "t \<in> Ma__"   | T :: "('fun, 'var) Term.term list" and f :: "'fun" and M :: "('fun, 'var) Term.term set" where "a1 = M" and "a2 = Fun f T" and "length T = arity f" and "public f" and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t"
cases: fixes x :: "'b aexp" obtains l2 :: "value" where "x = L l2"   | v2 :: "'b" where "x = V v2"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Plus e1__ e2__"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Minus e1a__ e2a__"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Times e1 e2"
path_prefix: fixes ds1 :: "bool list"   and ds2 :: "bool list"   and T :: "tree" assumes "path (ds1 @ ds2) T" shows "path ds1 T"
\<tau>move2_ThrowD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (throw e) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
runit_eqI: fixes f :: "'d"   and a :: "'d" assumes "\<guillemotleft>f : a \<otimes>\<^sub>D \<I>\<^sub>D \<rightarrow>\<^sub>D a\<guillemotright>"   and "f \<otimes>\<^sub>D \<I>\<^sub>D = (a \<otimes>\<^sub>D \<iota>\<^sub>D) \<cdot>\<^sub>D \<a>\<^sub>D[a, \<I>\<^sub>D, \<I>\<^sub>D]" shows "f = \<r>\<^sub>D[a]"
polynomial_function_inner: fixes g :: "'c \<Rightarrow> 'd"   and i :: "'d" assumes "polynomial_function g" shows "polynomial_function (\<lambda>x. g x \<bullet> i)"
word_le_0_iff: fixes x :: "'a word" shows "(x \<le> 0) = (x = 0)"
A1: shows "c_fst (c_fst key) mod 7 = 4"
nat_term_compare_POT: fixes to :: "'a nat_term_order" shows "nat_term_compare (POT to) = pot_comp (nat_term_compare to)"
val_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = val" shows "nsqn rt ip = sqn rt ip"
numeral_times_minus_swap: fixes w :: "num"   and x :: "'b" shows "numeral w * - x = x * - numeral w"
sub_casts: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and T' :: "ty"   and T :: "ty" assumes "wf_prog wf_md P"   and "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor>"   and "is_type P T"   and "P \<turnstile> T' \<le> T" shows "\<exists>v'. P \<turnstile> T casts v to v' "
that: fixes n' :: "'node"   and vs :: "'val list"   and v' :: "'val" assumes "n' \<in> set (successors g n)"   and "(n, v) \<in> set (zip (predecessors g n') vs)"   and "phis g (n', v') = Some vs" shows "thesis"
pre_newman_def: fixes s :: "complex" shows "pre_newman s = (\<Sum>p. if prime p then B p s else 0)"
new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'l" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
vertex_in_separator_2: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set"   and s :: "'g"   and t :: "'g"   and v :: "'e" assumes "TreeDecomposition G T bag"   and "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<in> TreeDecomposition.left_part T bag s t"   and "v \<in> TreeDecomposition.left_part T bag t s" shows "v \<in> bag t"
rel_FGco_def: fixes L1 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Co1 :: "'w \<Rightarrow> 'aj \<Rightarrow> bool"   and Co2 :: "'y \<Rightarrow> 'al \<Rightarrow> bool"   and Co3 :: "'z \<Rightarrow> 'am \<Rightarrow> bool"   and Co4 :: "'aa \<Rightarrow> 'an \<Rightarrow> bool"   and Co5 :: "'x \<Rightarrow> 'ak \<Rightarrow> bool"   and Co6 :: "'ae \<Rightarrow> 'aq \<Rightarrow> bool"   and Contra1 :: "'ab \<Rightarrow> 'ao \<Rightarrow> bool"   and Contra2 :: "'ac \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra3 :: "'af \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra4 :: "'ag \<Rightarrow> 'as \<Rightarrow> bool" shows "rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 = rel_F L1 Co1 Co5 (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) Co3 Co6 Contra1 Contra3 Contra4"
fls_base_factor_idem: fixes f :: "'a fls" shows "fls_base_factor (fls_base_factor f) = fls_base_factor f"
uint64_simps_3: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x - y) = (int_of_uint64 x - int_of_uint64 y) mod 18446744073709551616"
Qs: shows "set C \<subseteq> \<B>\<^sub>A s \<and> distinct C \<and> (\<forall>a\<in>set C. is_leq_ns (f (atom_var a) *R nsc_of_atom a) \<and> f (atom_var a) \<noteq> 0) \<and> (\<Sum>a\<leftarrow>C. lec_of_nsc (f (atom_var a) *R nsc_of_atom a)) = Le_Constraint Leq_Rel p c \<and> c < (0::'a) \<and> (\<forall>v. v \<Turnstile>\<^sub>t \<T> s \<longrightarrow> p \<lbrace> v \<rbrace> = (0::'a))"
prime_harmonic_lower': fixes n :: "nat" assumes "2 \<le> n" shows "ln (ln (real n + 1)) - ln (5 / 3) \<le> (\<Sum>p\<leftarrow>primes_upto n. 1 / real p)"
not_in_kD_nsqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<notin> kD rt" shows "nsqn rt dip = 0"
hom_add: fixes x :: "'b"   and y :: "'b" shows "to_fract (x + y) = to_fract x + to_fract y"
sds_automorphism: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R"   and "image_mset (map ((`) \<sigma>)) (anonymous_profile R) = anonymous_profile R" shows "map_pmf \<sigma> (sds R) = sds R"
inf_bot_left: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and x :: "'a" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf bot x = bot"
H_def: fixes A :: "'b set" shows "H A = map_pmf (\<lambda>f. Max (f ` A \<union> {0})) (SL A)"
disjE: fixes P :: "bool"   and Q :: "bool" assumes "P \<or> Q" obtains "P"   | "Q"
Cast: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned"   and T :: "ty" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>Cast T e\<rangle>\<guillemotright> A"
assms_1: shows "(f has_contour_integral i) (shiftpath a g)"
iso_unit_2: shows "\<guillemotleft>\<iota> : a \<star> a \<Rightarrow>\<^sub>S a\<guillemotright>"
wellformed_mk_symmetric: shows "pair_wf_digraph (mk_symmetric (with_proj (mk_symmetric G)))"
app2: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B \<rightarrow>\<beta> B'" shows "App A B \<rightarrow>\<beta> App A B'"
frrestrictionD_1: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A" shows "b \<in>\<^sub>\<circ> A"
dom_char: fixes f :: "'a" shows "Right_a.dom f = (if Right_a.arr f then local.dom f else null)"
set_offending_flows_def: fixes G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "set_offending_flows G nP = {F. F \<subseteq> edges G \<and> is_offending_flows_min_set F G nP}"
sq_norm_row_sylvester_mat2: fixes g :: "'a poly"   and i :: "nat"   and f :: "'a poly" assumes "degree g \<le> i"   and "i < degree f + degree g" shows "\<parallel>row (sylvester_mat f g) i\<parallel>\<^sup>2 = \<parallel>g\<parallel>\<^sup>2"
AndL2_in_SNa: fixes M :: "trm"   and x :: "name"   and y :: "name" assumes "SNa M" shows "SNa (AndL2 x.M y)"
set_Keys_to_list: fixes ps :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "set (punit.Keys_to_list ps) = Keys (set ps)"
len_s'_A: fixes n :: "nat"   and m :: "nat" shows "length (s'_A n m) = length init"
set_concat_len: fixes us :: "'b list list"   and G :: "'b list set"   and u :: "'b list" assumes "us \<in> lists G\<^sub>+"   and "1 < \<^bold>|us\<^bold>|"   and "u \<in> set us" shows "\<^bold>|u\<^bold>| < \<^bold>|concat us\<^bold>|"
minus_pCons: fixes a :: "'a"   and p :: "'a poly" shows "- pCons a p = pCons (- a) (- p)"
typeof_addr_hext_mono: fixes h :: "'heap"   and h' :: "'heap"   and a :: "'addr"   and hT :: "htype" assumes "h \<unlhd> h'"   and "typeof_addr h a = \<lfloor>hT\<rfloor>" shows "typeof_addr h' a = \<lfloor>hT\<rfloor>"
cfg'_def: fixes c :: "cmd"   and a :: "action"   and c' :: "cmd" shows "cfg' c a c' \<equiv> cfg c (Inl a) c'"
Cons_eq_map_D: fixes x :: "'b"   and xs :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and ys :: "'c list" assumes "x # xs = map f ys" shows "\<exists>z zs. ys = z # zs \<and> x = f z \<and> xs = map f zs"
scons_prefix_scons: fixes x :: "'a"   and xs :: "[:'a:]"   and y :: "'a"   and ys :: "[:'a:]" shows "(prefix\<cdot>(x :# xs)\<cdot>(y :# ys) = TT) = (eq\<cdot>x\<cdot>y = TT \<and> prefix\<cdot>xs\<cdot>ys = TT)"
square_free_factorization_prod_list: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)" shows "p = Polynomial.smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"
inInter_alt: fixes r :: "'c rexp"   and s :: "'c rexp" shows "inInter r s = (let X = toplevel_inters (rexp.Inter r s) - {Full} in INTERSECT (sorted_list_of_set (if Zero \<in> X then {Zero} else X)))"
Sigma_empty1: fixes B :: "'a \<Rightarrow> 'b set" shows "Sigma {} B = {}"
aadd_two_negg: fixes a :: "ant"   and b :: "ant" assumes "a < 0"   and "b < 0" shows "a + b < 0"
cV: fixes n :: "nat" shows "carrier (module_vec TYPE('b) n) = carrier_vec n"
trancl_unfold: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>+ = r \<union> r\<^sup>+ O r"
bij: shows "bij_betw ((div) a) {d. d dvd a} {d. d dvd a}"
no_props: shows "\<forall>no\<in>set prx \<union> set sfx. no \<noteq> Null \<and> (low no = Null) = (high no = Null) \<and> (isLeaf_pt p low high \<longrightarrow> isLeaf_pt no low high) \<and> var no = var p"
rep_eq: fixes x :: "('d, 'c) generator" shows "lgenerator (llist_of_trans x) = generator x"
assms: shows "(k, \<gamma>) \<in> boundary diamond_cube_gen"
mkarr_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C" shows "free_monoidal_category.mkarr C t = {u. (monoidal_language.Arr C t \<and> monoidal_language.Arr C u \<and> monoidal_language.Dom C t = monoidal_language.Dom C u \<and> monoidal_language.Cod C t = monoidal_language.Cod C u) \<and> monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u}"
single_addition_zero: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "0\<Odot>a\<^bsub>f,i,z\<^esub> = z"
nat_intv_frac_gt0: fixes c :: "nat"   and x :: "real" assumes "real c < x"   and "x < real (c + 1)" shows "0 < frac x"
simps_1: fixes nothing :: "unit"   and more :: "'m" shows "RObject.nothing \<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = nothing"
ha: shows "h a = \<lfloor>(C, fs)\<rfloor>"
exec_strip_to_exec: fixes F :: "'f set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
eventually_uniformity_metric: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and P :: "'c \<times> 'c \<Rightarrow> bool" assumes "class.uniformity_dist dist uniformity" shows "eventually P uniformity = (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))"
subseq_map: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<preceq> ys" shows "map f xs \<preceq> map f ys"
d: fixes ia :: "nat" assumes "ia \<le> m" shows "d fs' ia = d fs ia"
induct_step: fixes f :: "('b, 'c) urel"   and st :: "'a" assumes "(Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> f"   and "\<mu> B = f" shows "(Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B f"
simple_refine: shows "a.c.gen_dfs \<le> \<Down> (\<langle>ES\<rangle>simple_state_rel) a.a.it_dfs"
irreflp_onI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>a. a \<in> A \<Longrightarrow> \<not> P a a" shows "irreflp_on P A"
sat\<^sub>1\<sigma>: shows "eval\<^sub>c F G (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>)"
Z_def: shows "Z = {X. \<forall>i. 0 \<le> i \<and> i \<le> n - 1 \<longrightarrow> X i = x !! i}"
tt_ge_keys: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "\<And>u. u \<in> keys p \<Longrightarrow> v \<preceq>\<^sub>t u"   and "p \<noteq> 0" shows "v \<preceq>\<^sub>t tt p"
hyps: fixes xs :: "'a llist" assumes "llength xs = enat n" shows "lfinite xs"
assms_4: shows "\<lbrace>p \<cdot> !b\<rbrace>y\<lbrace>q\<rbrace>"
E: shows "nths xs A = nths (take i xs) (A \<inter> {..<i}) @ xs ! i # nths (drop (Suc i) xs) {j. Suc j + i \<in> A}"
simps_1: fixes f :: "'a fps"   and a :: "'a" shows "fps_right_inverse_constructor f a 0 = a"
IH: assumes "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>s\<in>trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t S. \<exists>X. set X \<subseteq> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<and> t = s \<cdot> rm_vars (set X) \<delta>"
lcm_A: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
cancel_times_right: fixes b :: "'a N"   and a :: "'a N"   and c :: "'a N" assumes "b * a = c * a" shows "b = c"
cross_ratio_correct_def: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" shows "cross_ratio_correct p q r s \<equiv> proj2_set_Col {p, q, r, s} \<and> p \<noteq> q \<and> r \<noteq> p \<and> s \<noteq> p \<and> r \<noteq> q"
prv_psubst_scnj: fixes F :: "'fmla set"   and txs :: "('trm \<times> 'var) list" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "prv (eqv (psubst (scnj F) txs) (scnj ((\<lambda>\<phi>. psubst \<phi> txs) ` F)))"
insertI2: fixes a :: "'b"   and B :: "'b set"   and b :: "'b" assumes "a \<in> B" shows "a \<in> b \<triangleright> B"
bit_eq_iff: fixes a :: "bit"   and b :: "bit" shows "(a = b) = (even a = even b)"
set_mult_closed: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#>\<^bsub>add_monoid G\<^esub> K \<subseteq> carrier G"
assms_6: shows "A\<^sub>P\<^sub>Q \<sharp>* Q"
next_recv: fixes c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and msg :: "'c"   and c' :: "('a, 'b, 'c) configuration" shows "c \<turnstile> Recv i p q s s' msg \<mapsto> c' = (can_occur (Recv i p q s s' msg) c \<and> (states c p = s \<and> states c' p = s') \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c i = Msg msg # msgs c' i \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>r. ps c r = ps c' r) \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') \<and> (if snd (cs c i) = Recording then cs c' i = (fst (cs c i) @ [msg], Recording) else cs c i = cs c' i))"
combine_points_index: fixes ps :: "'a set" shows "\<B>\<^sup>+ index ps = \<B> index ps + \<B>' index ps"
pw_leI': fixes S' :: "'b nres"   and S :: "'b nres" assumes "nofail S' \<Longrightarrow> nofail S"   and "\<And>x. \<lbrakk>nofail S'; inres S x\<rbrakk> \<Longrightarrow> inres S' x" shows "S \<le> S'"
inj_on_nat_permutes: fixes f :: "nat \<Rightarrow> nat"   and S :: "nat set" assumes "inj_on f S"   and "f \<in> S \<rightarrow> S"   and "finite S"   and "\<And>i. i \<notin> S \<Longrightarrow> f i = i" shows "f permutes S"
cut_greater_mem_iff: fixes x :: "'b"   and I :: "'b set"   and t :: "'b" shows "(x \<in> I \<down>> t) = (x \<in> I \<and> t < x)"
epi_implies_arr: fixes f :: "'b" assumes "D.epi f" shows "D.arr f"
scale: fixes r :: "'a"   and b :: "'b" shows "construct B f (r *a b) = r *b construct B f b"
rightreg_scalar_multD1: fixes x :: "'r"   and r :: "'r"   and f :: "'r \<Rightarrow> 'b" assumes "x \<in> R" shows "(r \<currency> f) x = f (x * r)"
image_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
compI_2: fixes f :: "'a"   and g :: "'a" assumes "f \<in> Ar"   and "g \<in> Ar"   and "Cod f = Dom g" shows "Dom (g \<bullet> f) = Dom f"
circ_back_loop_prefixpoint: fixes z :: "'a"   and y :: "'a" shows "z * y\<^sup>\<circ> * y \<squnion> z \<le> z * y\<^sup>\<circ>"
annil: fixes x :: "'a" shows "(0::'a) \<cdot> x = (0::'a)"
cmp_def: fixes \<mu>\<nu> :: "'b \<times> 'b" shows "FoH.cmp \<mu>\<nu> = (if B.VV.arr \<mu>\<nu> then F (H (fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)) \<cdot>\<^sub>D F (\<Phi>\<^sub>H (B.VV.dom \<mu>\<nu>)) \<cdot>\<^sub>D \<Phi>\<^sub>F (H (B.dom (fst \<mu>\<nu>)), H (B.dom (snd \<mu>\<nu>))) else D.null)"
product_suite_soundness: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and m :: "nat" assumes "well_formed M1"   and "well_formed M2"   and "observable M1"   and "observable M2"   and "inputs M2 = inputs M1"   and "|M1| \<le> m" shows "\<not> M1 \<preceq> M2 \<longrightarrow> \<not> M1 \<preceq>\<lbrakk>{xs. set xs \<subseteq> inputs M2 \<and> length xs \<le> |M2| * m}\<rbrakk> M2"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "flatten (=) (insert x A) (flatten (insort x) A)"
a: shows "a \<in> carrier (reduced_homology_group (int p) (nsphere p))"
reachable_neq_reachable1: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w"   and "v \<noteq> w" shows "v \<rightarrow>\<^sup>+ w"
H_m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> H"   and "y \<in> H" shows "x \<otimes> y \<in> H"
simps: fixes s :: "name"   and v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and k :: "name" assumes "atom s \<sharp> (v, i, t, u, k)"   and "atom k \<sharp> (v, i, t, u)" shows "AbstTermP v i t u = OrdP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v (Q_Ind i) t u (Var s) (Var k)))"
wf_if_measure_in_wf: fixes R :: "('a \<times> 'a) set"   and S :: "('b \<times> 'b) set"   and \<nu> :: "'b \<Rightarrow> 'a" assumes "wf R"   and "\<And>a b. (a, b) \<in> S \<Longrightarrow> (\<nu> a, \<nu> b) \<in> R" shows "wf S"
ksimplex_card: fixes p :: "nat"   and n :: "nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "ksimplex p n s" shows "card s = Suc n"
is_empty_default: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "\<not> Mapping.is_empty (Mapping.default k v m)"
dist_commute: fixes x :: "'a"   and y :: "'a" shows "dist x y = dist y x"
lin: shows "class.linorder (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
Ne_def: shows "Ne = (\<lambda>e. SOME N. ennreal (1 - e) < emeasure M {x \<in> space M. \<forall>B\<ge>N. real (card ({n. \<exists>l\<in>{N..n}. u n x - u (n - l) x \<le> - e * real l} \<inter> {..<B})) < e * real B})"
span_span: fixes A :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "extended_ord.punit.pmdl (extended_ord.punit.pmdl A) = extended_ord.punit.pmdl A"