hcju/isabelleps huggingface dataset:
intersect_mult_set_eq_block: fixes blv :: "'a set"   and p :: "'a" assumes "blv \<in># \<B>" shows "(p \<in># (\<Sum>bl\<in>#remove1_mset blv \<B>. mset_set (bl \<inter> blv))) = (p \<in> blv)"
int_pow_mult: fixes x :: "nat \<Rightarrow> 'a"   and i :: "int"   and j :: "int" assumes "x \<in> carrier P" shows "[(i + j)] \<cdot>\<^bsub>P\<^esub> x = [i] \<cdot>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> [j] \<cdot>\<^bsub>P\<^esub> x"
dual_ball: fixes A :: "'c set"   and P :: "'c dual \<Rightarrow> bool" shows "(\<forall>x\<in>A. P (dual x)) = (\<forall>x'\<in>dual ` A. P x')"
lead_coeff_def: fixes p :: "'a mpoly" shows "Symmetric_Polynomials.lead_coeff p = MPoly_Type.coeff p (lead_monom p)"
continuous_on_Pair: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. (f x, g x))"
J: shows "map the (filter (\<lambda>w. \<exists>y. w = Some y \<and> key y = k) (fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma)) = filter (\<lambda>y. k = key y) xs"
ipoly_complex_interval: fixes x :: "complex"   and X :: "complex_interval"   and p :: "int poly" assumes "x \<in>\<^sub>c X" shows "poly (complex_of_int_poly p) x \<in>\<^sub>c ipoly_complex_interval p X"
extf_ext_snoc: fixes f :: "'s" shows "ext_snoc (extf f)"
intro: fixes Y :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and cl :: "'b potype" assumes "Y \<subseteq> fix f (pset cl)" shows "Tarski_axioms cl f Y"
bet2_lta__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and D' :: "'p" assumes "A B C LtA D E F"   and "Bet A B A'"   and "A' \<noteq> B"   and "Bet D E D'"   and "D' \<noteq> E" shows "D' E F LtA A' B C"
1: fixes X :: "'a list" shows "v \<notin> subst_domain (rm_vars (set X) \<delta>)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data" assumes "three_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi>"   and "identity_arrow_of_spans C \<mu>"   and "identity_arrow_of_spans C \<nu>"   and "identity_arrow_of_spans C \<pi>"   and "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>"   and "two_composable_identity_arrows_of_spans C prj0 prj1 \<nu> \<pi>" shows "three_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi>"
isLastAssertedLiteral_def: fixes literal :: "Literal"   and clause :: "Literal list"   and valuation :: "Literal list" shows "isLastAssertedLiteral literal clause valuation \<equiv> literal el clause \<and> literalTrue literal valuation \<and> (\<forall>literal'. literal' el clause \<and> literal' \<noteq> literal \<longrightarrow> \<not> precedes literal literal' valuation)"
maddux1a': fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a \<cdot> (x \<sqinter> - (a \<rhd> y)) \<le> - y"
wlsEnv_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "wlsEnv rho \<equiv> (\<forall>ys. liftAll (wls (asSort ys)) (rho ys)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|)"
\<mu>LTL_def: shows "\<mu>LTL = {\<phi>. is_\<mu>LTL \<phi>}"
timpl_closure_idem: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "timpl_closure_set (timpl_closure t TI) TI = timpl_closure t TI"
contr: assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Stuck" shows "False"
prec_t: shows "precondition (next_state t execs2) (next_action t execs2)"
commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U" shows "local.min a b = local.min b a"
ekMeasure_upper_bound: shows "ek_analysis_defs.ekMeasure (cf_of (\<lambda>_. 0::'capacity)) s t < 2 * card V * card E + card V"
cis_cnj: fixes t :: "real" shows "cnj (cis t) = cis (- t)"
cases: fixes x :: "'c Tree" obtains v :: "'c" and l :: "'c Tree" and r :: "'c Tree" where "x = T v l r"   | "x = E"
minimalize_offending_overapprox_spec: shows "TopoS_Interface_impl.minimalize_offending_overapprox (implc_sinvar m_impl) (edgesL G) [] G = TopoS_withOffendingFlows.minimalize_offending_overapprox (c_sinvar m_spec) (edgesL G) [] (list_graph_to_graph G)"
reachable_union_sym: fixes r :: "('b \<times> 'b) set"   and a :: "'b ref"   and aS :: "'b ref set" shows "reachable r (insert a aS) = r\<^sup>* `` addrs {a} \<union> reachable r aS"
mat_kernel_carrier: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "mat_kernel A \<subseteq> carrier_vec nc"
seqsimilarI: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "first s = first t"   and "nextsuffix s \<approx> nextsuffix t" shows "s \<approx> t"
rec_4: assumes "Keys (set (snd (sym_preproc_addnew gs vs' fs v))) = set (ks @ [v]) \<union> set (fst (sym_preproc_addnew gs vs' fs v))" shows "set (fst (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))) = Keys (set (snd (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))))"
numeral_le_one_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus n) one = (n \<le> num.One)"
Sbis_pres_discr_R: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "discr c"   and "c \<approx>s d" shows "discr d"
poisson_def: fixes rate :: "val" shows "poisson rate = density (stock_measure INTEG) (poisson_density' (extract_real rate) \<circ> extract_int)"
assms_8: fixes p' :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and q' :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(p', q') \<in> snd ` set (apply_ncrit crit data gs bs hs ps)"   and "fst p' \<noteq> 0"   and "fst q' \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"
possible_steps_2_coin: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "length i = 1" shows "possible_steps drinks2 2 r STR ''coin'' i = {|(2, coin)|}"
src_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>C (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
product_mix_pmf_prob_distrib: fixes a :: "real"   and b :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "a \<in> {0..1}"   and "b \<in> {0..1}" shows "mix_pmf a (mix_pmf b p q) q = mix_pmf (a * b) p q"
kbo_supt_one: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and bef :: "('f, 'b) Term.term list"   and aft :: "('f, 'b) Term.term list" assumes "NS s t" shows "S (Fun f (bef @ s # aft)) t"
sorted_ssort': fixes l :: "'b"   and sl :: "'a list" assumes "inv l"   and "sorted sl \<and> (\<forall>x\<in>local.set l. \<forall>y\<in>list.set sl. x \<le> y)" shows "sorted (ssort' l sl)"
cases_scheme: fixes r :: "('c, 'd) alpha_rp_scheme" obtains ok :: "bool" and wait :: "bool" and tr :: "'c list" and ref :: "'c set" and more :: "'d" where "r = \<lparr>ok = ok, wait = wait, tr = tr, ref = ref, \<dots> = more\<rparr>"
trivial_in_allVars: fixes g :: "'g"   and v :: "'val" assumes "trivial g v" shows "v \<in> allVars g"
ode_d1_has_derivative_safeI: fixes x :: "'c" assumes "x \<in> Csafe" shows "(ode_d1 has_derivative ode_d (Suc 0) x) (at x)"
inject: fixes x :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf"   and ya :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" shows "(GPV x = GPV ya) = (x = ya)"
geodesic_segment_homeo_interval: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "{0..dist x y} homeomorphic G"
subspaceI: fixes S :: "'d set" assumes "(0::'d) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c *\<^sub>C x \<in> S" shows "csubspace S"
fw_shortest_path: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "linordered_ab_monoid_add.D plus less_eq m i j k = linordered_ab_monoid_add.fw plus less_eq m n k i j"
state_set_thm: fixes s1 :: "'b list"   and s2 :: "'b list" shows "(s1 \<in> state_set s2) = (prefix s1 s2 \<and> s1 \<noteq> [])"
cases: fixes x :: "('d, 'e) stateful_strand_step" obtains t :: "('d, 'e) Term.term" where "x = send\<langle>t__\<rangle>"   | t :: "('d, 'e) Term.term" where "x = receive\<langle>ta__\<rangle>"   | uu_ :: "poscheckvariant" and t :: "('d, 'e) Term.term" and t' :: "('d, 'e) Term.term" where "x = \<langle>uu_: tb__ \<doteq> t'__\<rangle>"   | t :: "('d, 'e) Term.term" and t' :: "('d, 'e) Term.term" where "x = insert\<langle>tc__,t'a__\<rangle>"   | t :: "('d, 'e) Term.term" and t' :: "('d, 'e) Term.term" where "x = delete\<langle>td__,t'b__\<rangle>"   | uv_ :: "poscheckvariant" and t :: "('d, 'e) Term.term" and t' :: "('d, 'e) Term.term" where "x = \<langle>uv_: t \<in> t'\<rangle>"   | uw_ :: "'e list" and F :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" and F' :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" where "x = \<forall>uw_\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle>"
WT: shows "\<I> \<turnstile>g d \<surd>"
update_convs_7: fixes safe' :: "(room \<Rightarrow> bool) \<Rightarrow> room \<Rightarrow> bool"   and owns :: "room \<Rightarrow> guest option"   and currk :: "room \<Rightarrow> key"   and issued :: "key set"   and cards :: "guest \<Rightarrow> (key \<times> key) set"   and roomk :: "room \<Rightarrow> key"   and isin :: "room \<Rightarrow> guest set"   and safe :: "room \<Rightarrow> bool"   and more :: "'a" shows "safe_update safe' \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin, safe = safe, \<dots> = more\<rparr> = \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin, safe = safe' safe, \<dots> = more\<rparr>"
id_is_unitary: fixes n :: "nat" shows "unitary (1\<^sub>m n)"
ring_simprules_7: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
mulDisj: fixes S :: "'a set"   and T :: "'a set set"   and mx :: "'a set" assumes "mul_closed_set R S"   and "1\<^sub>r \<in> S"   and "\<zero> \<notin> S"   and "T = {I. ideal R I \<and> S \<inter> I = {}}"   and "maximal_set T mx" shows "prime_ideal R mx"
contour_integral_part_circlepath_eq: fixes a :: "real"   and b :: "real"   and c :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex" assumes "a < b" shows "contour_integral (part_circlepath c r a b) f = integral {a..b} (\<lambda>t. f (c + complex_of_real r * cis t) * complex_of_real r * \<i> * cis t)"
standard_uniqueness: fixes W :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and C :: "'b set" assumes "ChamberComplexMorphism W X f"   and "ChamberComplexMorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C"   and "\<And>Cs. SimplicialComplex.min_maxsimpchain W (C # Cs) \<Longrightarrow> pgallery (f \<Turnstile> (C # Cs))"   and "\<And>Cs. SimplicialComplex.min_maxsimpchain W (C # Cs) \<Longrightarrow> pgallery (g \<Turnstile> (C # Cs))" shows "fun_eq_on f g (\<Union> W)"
rcis_inverse: fixes r :: "real"   and a :: "real" shows "inverse (rcis r a) = rcis (1 / r) (- a)"
count_filter_zmset: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a zmultiset"   and a :: "'a" shows "zcount (filter_zmset P M) a = (if P a then zcount M a else 0)"
normalize_quot_div_unit_left: fixes u :: "'b"   and x :: "'b"   and y :: "'b" assumes "is_unit u" shows "normalize_quot (x div u, y) = (fst (normalize_quot (x, y)) div u, snd (normalize_quot (x, y)))"
sorted_list_of_set_map_mono: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "\<forall>n m. n < m \<longrightarrow> f n < f m"   and "finite A" shows "map f (sorted_list_of_set A) = sorted_list_of_set (f ` A)"
mset_subset_eq_multiset_union_diff_commute: fixes B :: "'c multiset"   and A :: "'c multiset"   and C :: "'c multiset" assumes "B \<subseteq># A" shows "A - B + C = A + C - B"
differing_vars_lists_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat" shows "differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i = differing_vars mem\<^sub>1 (fst (mems ! i)) \<union> differing_vars mem\<^sub>2 (snd (mems ! i))"
monom_mult_one_left: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult (1::'d) (0::'a) p = p"
that: shows "prove_nonneg prnt i p (slp_of_fas [fa]) YSS"
l_comb_mem_linear_span: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H" shows "l_comb R M n s f \<in> linear_span R M A H"
solution_0_snd_nat_iff: fixes a :: "nat" shows "solution (a, 0) = (a = 1)"
changes_map_sgn_eq: fixes xs :: "'a list" shows "changes xs = changes (map sgn xs)"
strengthen_pre: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile>\<^sub>t {P} c {Q}" shows "\<turnstile>\<^sub>t {P'} c {Q}"
init_inv_gc: shows "\<forall>s. initial_state \<lparr>PGMs = gc_coms, INIT = gc_system_init, FAIR = \<lambda>s. True\<rparr> s \<longrightarrow> gc.invsL \<lparr>GST = s, HST = []\<rparr>"
fresh_perm_app: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'a" assumes "a \<sharp> pi"   and "a \<sharp> x" shows "a \<sharp> pi \<bullet> x"
leadsto_infinite: fixes F :: "'a \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>F \<and> (F \<leadsto> G) \<longrightarrow> \<box>\<diamond>G"
boolLess_def: shows "boolLess = {(True, False)}"
basis_wf_insert_ln_1: fixes b :: "real \<Rightarrow> real" assumes "basis_wf [b]" shows "basis_wf [\<lambda>x. ln (b x)]"
norm_notMT: fixes p :: "((int \<times> int) set set, int) Combinators" assumes "DenyAll \<in> set (policy2list p)" shows "FWNormalisationCore.normalize p \<noteq> []"
assms_2: shows "D + {#L'#} \<in> CC"
count_prime_factorization_prime: fixes p :: "'a"   and x :: "'a" assumes "prime p" shows "count (prime_factorization x) p = multiplicity p x"
ereal_zero_mult: fixes a :: "ereal" shows "0 * a = 0"
expandDepth: fixes P :: "pi"   and F :: "pi set" assumes "(P, F) \<in> sumComposeSet"   and "F \<noteq> {}" shows "\<exists>Q\<in>F. depth P \<le> depth Q \<and> (\<forall>R\<in>F. depth R \<le> depth Q)"
map\<^sub>0_in_hhom: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : src\<^sub>D (F a) \<rightarrow>\<^sub>D src\<^sub>D (G a)\<guillemotright>"
gpd: shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
fun_relD1: fixes f :: "'f \<Rightarrow> 'g"   and f' :: "'h \<Rightarrow> 'i"   and Ra :: "('f \<times> 'h) set"   and Rr :: "('g \<times> 'i) set"   and x :: "'f"   and r :: "'g" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
ipoly_poly_compose: fixes p :: "int poly"   and q :: "int poly"   and x :: "'b" shows "ipoly (p \<circ>\<^sub>p q) x = ipoly p (ipoly q x)"
BseqE: fixes X :: "nat \<Rightarrow> 'b" assumes "Bseq X" obtains K :: "real" where "0 < K" and "\<forall>n. norm (X n) \<le> K"
Inf_aform'_def: fixes p :: "nat"   and X :: "'a \<times> 'a pdevs" shows "Inf_aform' p X = eucl_truncate_down p (fst X - Radius' p X)"
pos_imp_zdiv_pos_iff: fixes k :: "int"   and i :: "int" assumes "0 < k" shows "(0 < i div k) = (k \<le> i)"
prime_field_def: fixes p :: "int" shows "prime_field TYPE('a) p \<equiv> mod_ring_locale TYPE('a) p"
set_eq_iff: fixes A :: "'d set"   and B :: "'d set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
trace_callee_eq_run_gpv: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and I1 :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and I2 :: "'g \<Rightarrow> bool"   and gpv :: "('h, 'e, 'f) gpv" assumes "trace_callee_eq callee1 callee2 A p q"   and "callee_invariant_on callee1 I1 \<I>"   and "callee_invariant_on callee2 I2 \<I>"   and "\<I> \<turnstile>g gpv \<surd>"   and "outs_gpv \<I> gpv \<subseteq> A"   and "lossless_spmf p"   and "lossless_spmf q"   and "\<forall>x\<in>set_spmf p. I1 x"   and "\<forall>y\<in>set_spmf q. I2 y" shows "p \<bind> run_gpv callee1 gpv = q \<bind> run_gpv callee2 gpv"
order_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "order r = ((\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> x) \<and> (\<forall>x y. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> x \<longrightarrow> x = y) \<and> (\<forall>x y z. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> z))"
tgt_V: shows "target (vs || trV) q1 = q2"
max_Var_floatariths_subst_floatarith_le: fixes fas :: "floatarith list"   and xs :: "floatarith list"   and z :: "nat" assumes "max_Var_floatariths fas \<le> length xs"   and "max_Var_floatariths xs \<le> z" shows "max_Var_floatariths (map (subst_floatarith ((!) xs)) fas) \<le> z"
negligible_Int: fixes s :: "'c set"   and t :: "'c set" assumes "negligible s \<or> negligible t" shows "negligible (s \<inter> t)"
restrict_relp_DomainpD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c" assumes "Domainp (R \<upharpoonleft> P \<otimes> Q) x" shows "Domainp R x \<and> P x"
nprv_insertShift2I: fixes \<phi>3 :: "'fmla"   and \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "nprv (insert \<phi>3 (insert \<phi>1 (insert \<phi>2 F))) \<psi>" shows "nprv (insert \<phi>1 (insert \<phi>2 (insert \<phi>3 F))) \<psi>"
test_bit_set: fixes w :: "'b word"   and n :: "nat"   and x :: "bool" shows "set_bit_class.set_bit w n x !! n = (n < size w \<and> x)"
cod_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> R.codomains f" shows "a = R.cod f"
natural_transformation_axioms: shows "natural_transformation A B G H \<tau>'"
composable_arrsI: fixes gf :: "V"   and g :: "V"   and f :: "V"   and \<CC> :: "V"   and b :: "V"   and c :: "V"   and a :: "V" assumes "gf = [g, f]\<^sub>\<circ>"   and "g : b \<mapsto>\<^bsub>\<CC>\<^esub> c"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "gf \<in>\<^sub>\<circ> composable_arrs \<CC>"
Refsum: shows "(\<lambda>x. Re (f x)) abs_summable_on A"
simps_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [] = True"
sprod_assoc: fixes a :: "'b"   and b :: "'b"   and m :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "m \<in> carrier M" shows "(a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<cdot>\<^sub>s m = a \<cdot>\<^sub>s (b \<cdot>\<^sub>s m)"
odd_of_bool_self: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "bool" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (zero_neq_one.of_bool one zero p)) = p"
LLL_invD_1: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
IH: fixes x2a :: "('a, 'b) Term.term" assumes "x2a \<in> set T"   and "f \<in> funs_term x2a" shows "\<exists>T. Fun f T \<sqsubseteq> x2a"
sgn_ell2_def: fixes x :: "'a ell2" shows "sgn x = x /\<^sub>R norm x"
length_hd_le_concat: fixes as :: "'a list list" assumes "as \<noteq> []" shows "length (hd as) \<le> length (concat as)"
principal_prod_principal: fixes A :: "'d set"   and B :: "'e set" shows "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
next_between: fixes f :: "face"   and v :: "nat"   and u :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "u \<in> \<V> f"   and "f \<bullet> v \<noteq> u" shows "f \<bullet> v \<in> set (between (vertices f) v u)"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "cmod (g a) \<le> 1 + cmod (f a) / 3"   and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (complex_of_real pi * g z)" shows "thesis"
list_of_dlist: fixes x :: "'a dlist" shows "list_of_dlist x \<in> {xs. distinct xs}"
Enc_keys_cleanI: fixes S :: "msg set" assumes "\<forall>X Y. Enc X Y \<in> parts S \<longrightarrow> Y \<in> range LtK \<union> payload" shows "Enc_keys_clean S"
terminatesI: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "\<And>s. s \<in> terminates_on g" shows "terminates g"
extract_matrix_def: fixes a :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "extract_matrix a m n = mat m n (\<lambda>(i, j). a (i * n + j))"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q"   and "\<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q" shows "Prop (P' \<parallel> Q)"
fresh_1: fixes a :: "vrs"   and x1 :: "tyvrs" shows "a \<sharp> Tvar x1 = a \<sharp> x1"
closed_components: fixes s :: "'a set"   and c :: "'a set" assumes "closed s"   and "c \<in> components s" shows "closed c"
less_le_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
hyp_port_eq: fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "f = subst (iSubst (tree_at t (hyp_port_path_for t is f))) (freshen (iAnnot (tree_at t (hyp_port_path_for t is f))) (labelsOut (iNodeOf (tree_at t (hyp_port_path_for t is f))) (hyp_port_h_for t is f)))"
comm_ring_ow_axioms: shows "comm_ring_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (-\<^sub>o\<^sub>w) uminus (*\<^sub>o\<^sub>w)"