hcju/isabelleps huggingface dataset:
connectedin_inter_frontier_of: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "connectedin X S"   and "S \<inter> T \<noteq> {}"   and "S - T \<noteq> {}" shows "S \<inter> X frontier_of T \<noteq> {}"
prj1_simps_arr: fixes f :: "'a"   and g :: "'a" shows "C.arr \<p>\<^sub>1[f, g] = C.cospan f g"
assms_3: shows "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops' \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops'"
ce_rel_lm_17: fixes x :: "nat"   and A :: "nat set" assumes "(c_fst x, c_snd x) \<in> ce_set_to_rel A" shows "x \<in> A"
Q_def: shows "Q \<equiv> trms\<^sub>l\<^sub>s\<^sub>t A"
mult_cancel_left1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times c b) = (c = zero \<or> b = one)"
Nmlize_Nml: fixes t :: "'a local.term" assumes "Nml t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = t"
ide_Ide: fixes a :: "'a1 \<times> 'a2" assumes "C1.ide (fst a) \<and> C2.ide (snd a)" shows "ide a"
AxiomsSet7orig_axioms: shows "AxiomsSet7orig source target (\<^bold>\<cdot>)"
typeof_xs: shows "list_all (\<lambda>x. x = None) (map typeof xs)"
ide_MkIde: fixes A :: "'obj" assumes "A \<in> Obj" shows "ide (MkIde A)"
Pj_ghom: fixes N :: "'a set" assumes "G \<triangleright> N" shows "Pj G N \<in> gHom G (G / N)"
hom_mult: fixes x :: "real poly"   and y :: "real poly" shows "map_poly complex_of_real (x * y) = map_poly complex_of_real x * map_poly complex_of_real y"
hsnd_def: fixes p :: "hf" shows "hsnd p \<equiv> THE y. \<exists>x. p = \<langle>x, y\<rangle>"
trace_cons: fixes conf' :: "'c"   and T :: "'c list"   and final :: "'c"   and conf :: "'c" assumes "trace conf' T final"   and "conf \<Rightarrow> conf'" shows "trace conf (conf' # T) final"
assms_2: shows "\<Psi> \<turnstile> \<Phi>"
reachableNodes_E_closed: fixes s :: "nat" shows "E `` reachableNodes s \<subseteq> reachableNodes s"
rec_list_Nil_imp: fixes f :: "'b list \<Rightarrow> 'c"   and f1 :: "'c"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'c \<Rightarrow> 'c" assumes "f = rec_list f1 f2" shows "f [] = f1"
act_def: fixes I :: "'b \<Rightarrow> bool" shows "act I \<equiv> \<lambda>_ _. I"
product_nn_integral_pair: fixes f :: "val \<Rightarrow> val \<Rightarrow> ennreal"   and x :: "nat"   and y :: "nat" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (stock_measure (\<Gamma> x) \<Otimes>\<^sub>M stock_measure (\<Gamma> y))"   and "x \<noteq> y" shows "\<integral>\<^sup>+ \<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>Pi\<^sub>M {x, y} (\<lambda>x. stock_measure (\<Gamma> x)) = \<integral>\<^sup>+ z. f (fst z) (snd z) \<partial>stock_measure (\<Gamma> x) \<Otimes>\<^sub>M stock_measure (\<Gamma> y)"
run_alt_writer: fixes m :: "('w, 'a, 'm) writerT"   and m' :: "('w, 'a, 'm) writerT" shows "run_writer (local.alt_writer m m') = alt (run_writer m) (run_writer m')"
divide_eq_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(divide b c = a) = (if c \<noteq> zero then b = times a c else a = zero)"
mu'_mu_i_im1: shows "gs2.\<mu> i (i - 1) = fs.gs.\<mu> i (i - 1) * \<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2 / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2"
strict_implies_order: fixes a :: "'c list"   and b :: "'c list" assumes "strict_prefix a b" shows "prefix a b"
LI_rel_def: shows "LI_rel \<equiv> {((xa, x), xc, xb). LI_relp xa x xc xb}"
hideit_def: fixes x :: "'a"   and y :: "'a" shows "hideit x y = Plus rexp.One (nodouble x y)"
mat_diag_mult_right: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "A \<in> carrier_mat nr n" shows "A * mat_diag n f = Matrix.mat nr n (\<lambda>(i, j). A $$ (i, j) * f j)"
zlfm_I: fixes p :: "fm"   and bbs :: "bool list"   and i :: "int"   and bs :: "int list" assumes "qfree p" shows "Ifm bbs (i # bs) (zlfm p) = Ifm bbs (i # bs) p \<and> iszlfm (zlfm p)"
exhaust: fixes y :: "finite_3" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"
subset_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'b set" assumes "inj_on f B"   and "order.greater_eq B A" shows "inj_on f A"
UNIF1_preserves_unifiers: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and \<tau> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "UNIF1 \<sigma> E E'"   and "\<tau> \<in> unifiers (set_mset E)" shows "\<sigma> \<circ>\<^sub>s \<tau> \<in> unifiers (set_mset E')"
simps_1: fixes x :: "'b equation" shows "atom (Pos x) = x"
content_eq_fold_coeffs: fixes p :: "'a poly" shows "content p = fold_coeffs gcd p (0::'a)"
pdoWk: fixes k :: "nat" shows "partial_density_operator (denote_while_n_iter M0 M1 DS k \<rho>)"
dom_exc_add_distr: fixes s :: "'c set"   and m\<^sub>1 :: "('c, 'd) fmap"   and m\<^sub>2 :: "('c, 'd) fmap" shows "s \<lhd>/ (m\<^sub>1 ++\<^sub>f m\<^sub>2) = s \<lhd>/ m\<^sub>1 ++\<^sub>f s \<lhd>/ m\<^sub>2"
spanning_forest_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "spanning_forest F G \<equiv> forest F \<and> maximally_connected F G \<and> subgraph F G"
consistent_add_instance: fixes S :: "('a, 'b) fm set"   and p :: "('a, 'b) fm"   and t :: "'a tm" assumes "consistent S"   and "\<^bold>\<forall> p \<in> S" shows "consistent ({p\<langle>t/0\<rangle>} \<union> S)"
complete_lattice_partial_function_definitions: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions less_eq Sup"
minus_cis: fixes x :: "real" shows "- cis x = cis (x + pi)"
add_eq_inf_sup: fixes a :: "'a"   and b :: "'a" shows "a + b = sup a b + inf a b"
Status_def: fixes HA :: "('a, 'b, 'c) hierauto"   and C :: "'a set"   and E :: "'b set"   and D :: "'c data" shows "Status HA C E D \<equiv> E \<subseteq> HAEvents HA \<and> IsConfSet (SAs HA) (CompFun HA) C \<and> Data.DataSpace (HAInitValue HA) = Data.DataSpace D"
lift_set_def: fixes i :: "nat"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "lift_set i A \<equiv> lift_map i ` A"
index_transpose_mat_3: fixes A :: "'a Matrix.mat" shows "dim_col A\<^sup>t = dim_row A"
PreFunctor_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "PreFunctor F \<equiv> ((\<forall>f g. f \<approx>>\<^bsub>CatDom F\<^esub> g \<longrightarrow> F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)) \<and> (\<forall>X. X \<in> obj\<^bsub>CatDom F\<^esub> \<longrightarrow> (\<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y))) \<and> Category (CatDom F) \<and> Category (CatCod F)"
plane_of_def: fixes sctn :: "'b sctn" shows "plane_of sctn = plane (normal sctn) (pstn sctn)"
span: shows "local.span (set (basis_completion v)) = carrier_vec n"
R_F: fixes N' :: "'a stream measure"   and M' :: "'a stream measure" assumes "(N', M') \<in> F" shows "R N' M'"
circle_right_top_edges_neq: shows "circle_right_edge \<noteq> circle_top_edge"
mod_ring_finite_field_ops32: fixes p :: "int" assumes "mod_ring_locale TYPE('a) p"   and "p \<le> 65535" shows "mod_ring_gen (finite_field_ops32 (uint32_of_int p)) (mod_ring_locale.mod_ring_rel32 p) p"
assms_1: shows "event_at (i, k) (Receive s m)"
simps: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b"   and f :: "'b poly"   and g :: "'b poly"   and c :: "'b" shows "subresultant_prs_main div_exp f g c = (let m = degree f; n = degree g; lf = lead_coeff f; lg = lead_coeff g; \<delta> = m - n; d = div_exp lg c \<delta>; h = pseudo_mod f g in if h = 0 then (g, d) else subresultant_prs_main div_exp g (sdiv_poly h ((- (1::'b)) ^ (\<delta> + 1) * lf * c ^ \<delta>)) d)"
Rat_cases: fixes q :: "rat" obtains a :: "int" and b :: "int" where "q = Fract a b" and "0 < b" and "coprime a b"
shiftl_word_transfer: shows "rel_fun pcr_word (rel_fun (=) pcr_word) (\<lambda>k n. push_bit n k) (<<)"
assms_2: fixes p :: "'a poly" assumes "\<And>a. P a \<Longrightarrow> poly p a \<noteq> (0::'a)" shows "Q p"
f2: shows "P \<turnstile> \<langle>e,(h, l |` W, sh)\<rangle> \<Rightarrow> \<langle>throw a,(h', l' |` W, sh')\<rangle>"
inj_on_mod_nat: fixes m :: "nat"   and a :: "nat" assumes "0 < m" shows "inj_on (\<lambda>x. x mod m) {a..<a + m}"
2: fixes A :: "'a tensor" assumes "A \<in> set (map (\<lambda>A. local.tensor_prod_otimes A B) As)" shows "dims A = ds @ dims B"
reduced_row_echelon_form_def': fixes A :: "(('b, 'c) vec, 'd) vec" shows "reduced_row_echelon_form A = ((\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)) \<and> (\<forall>i. \<not> is_zero_row i 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 i A \<and> \<not> is_zero_row (i + (1::'d)) A \<longrightarrow> (LEAST k. A $ i $ k \<noteq> (0::'b)) < (LEAST k. A $ (i + (1::'d)) $ k \<noteq> (0::'b))) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (0::'b))))"
vanishes_def: fixes X :: "nat \<Rightarrow> rat" shows "Real.vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
assms_1: shows "\<nexists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"
hop_count_zero_oip_dip_sip: fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg msg_zhops \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:4..PAodv-:5} \<union> {PRreq-:n |n. True} \<longrightarrow> hops \<xi> = 0 \<longrightarrow> oip \<xi> = sip \<xi>) \<and> (l \<in> {PAodv-:6..PAodv-:7} \<union> {PRrep-:n |n. True} \<longrightarrow> hops \<xi> = 0 \<longrightarrow> dip \<xi> = sip \<xi>))"
osum_wf: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "wf r"   and "wf r'" shows "wf (r +o r')"
expectation_gpv_cmult: fixes c :: "ennreal"   and fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal"   and gpv :: "('c, 'a, 'b) gpv" assumes "0 < c"   and "c \<noteq> \<top>" shows "c * expectation_gpv fail \<I> f gpv = expectation_gpv (c * fail) \<I> (\<lambda>x. c * f x) gpv"
mk_linear_orders_non_empty: fixes B :: "'a set"   and C :: "'a set" assumes "B \<subseteq> A" shows "\<exists>r. r \<in> local.mk_linear_orders C B"
assms_5: shows "A\<^sub>P \<sharp>* C"
C3: shows "args s = take (num_args t - Suc (length p)) (args t) @ drop (Suc (num_args t - Suc (length p))) (args t)"
semiring_norm_170: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "- numeral v * (numeral w * y) = - numeral (v * w) * y"
foldl_absorb0: fixes x :: "'b \<Rightarrow> bool"   and zs :: "('b \<Rightarrow> bool) list" shows "(x \<and>* foldl (\<and>*) \<box> zs) = foldl (\<and>*) x zs"
b: assumes "x \<noteq> y"   and "rel x y" shows "filter P (merge_wrt rel xs (y # ys)) = filter P xs"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_unary a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> \<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s \<and> num_args t = 1 \<and> (\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0) \<and> arg t \<ge>\<^sub>t s)"
2: shows "[a] @- v\<^sub>2 =\<^sub>I l @- u"
assms_1: shows "k-smooth_on (apply_chart c ` X) (f \<circ> inv_chart c)"
Liminf_state_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Sts :: "('b literal multiset set \<times> 'b literal multiset set \<times> 'b literal multiset set) llist" assumes "FO_resolution_prover S subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm" shows "FO_resolution_prover.Liminf_state Sts = (Liminf_llist (lmap FO_resolution_prover.N_of_state Sts), Liminf_llist (lmap FO_resolution_prover.P_of_state Sts), Liminf_llist (lmap FO_resolution_prover.Q_of_state Sts))"
mod_h_bot_iff_1: fixes b :: "bool"   and h :: "heap" shows "((h, {}) \<Turnstile> \<up> b) = b"
Keys_badI: fixes IK :: "msg set"   and Bad :: "agent set" assumes "IK \<inter> range LtK \<subseteq> range pubK \<union> priK ` Bad \<union> {shrK A B |A B. A \<in> Bad \<or> B \<in> Bad}"   and "range pubK \<subseteq> IK" shows "Keys_bad IK Bad"
ArityAnalysisLetSafeNoCard_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysisLetSafeNoCard Aexp Aheap \<equiv> ArityAnalysisLetSafe Aexp Aheap \<and> ArityAnalysisLetSafeNoCard_axioms Aheap"
parts_overlap_Fs: fixes F :: "i \<Rightarrow> bool" assumes "GEM1"   and "\<exists>x. F x" shows "\<forall>y. P y (\<sigma>x. F x) \<longrightarrow> (\<exists>z. F z \<and> O y z)"
ex_MAXIMUM: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>a\<in>A. Max (f ` A) = f a"
simps_1: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'c"   and w :: "'c"   and xs :: "'c list" assumes "Digraph G" shows "Digraph.first_edge_of_walk (v # w # xs) = (v, w)"
irreducible_uminus: fixes x :: "'a" shows "irreducible (- x) = irreducible x"
divides_mult_l: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "c \<otimes> a divides c \<otimes> b = a divides b"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cmp x y = Lt"   and "cmp y z = Lt" shows "cmp x z = Lt"
divide_lt: fixes c :: "rat"   and a :: "'a"   and b :: "'a" assumes "c *R a < b"   and "0 < c" shows "a < (1 / c) *R b"
openin_topology\<^sub>N: fixes N :: "'b quasinorm"   and U :: "'b set" shows "openin (topology\<^sub>N N) U = (\<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
IH: fixes ds :: "bool list"   and T :: "tree" assumes "path p (delete ds T)" shows "\<nexists>c cs. p = ds @ c # cs"
gcd_greatest_mult: fixes 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 gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and d :: "'b"   and b :: "'b" assumes "class.idom_gcd times one plus zero minus uminus gcd"   and "dvd.dvd times c (times a d)"   and "dvd.dvd times c (times b d)" shows "dvd.dvd times c (times (gcd a b) d)"
strt_lemma: fixes xs :: "nat list"   and vl :: "nat" assumes "length xs = vl" shows "rec_exec (rec_strt vl) xs = strt xs"
iterate_to_list_foldli: fixes xs :: "'c list" shows "iterate_to_list (foldli xs) = rev xs"
sel_2: fixes x2 :: "'b" shows "un_Prim \<^bold>\<langle>x2\<^bold>\<rangle> = x2"
mult_of_simps_2: fixes R :: "('b, 'c) ring_scheme" shows "(\<otimes>\<^bsub>mult_of R\<^esub>) = (\<otimes>\<^bsub>R\<^esub>)"
simps_2: fixes sys :: "'a \<Rightarrow> ('b, 'c) process_term"   and p :: "'d process" shows "isACT sys (PROC p) = isAction p"
trans_S_point: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "pre_order_pair S NS"   and "(x, y) \<in> S"   and "(y, z) \<in> S" shows "(x, z) \<in> S"
finite_dom: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" shows "finite (dom \<sigma>)"
cball_def: fixes x :: "'g"   and e :: "real" shows "cball x e = {y. dist x y \<le> e}"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and sep_size :: "nat"   and paths :: "'c list set"   and P_new :: "'c list" assumes "NoSmallSeparationsInduct G v0 v1 sep_size"   and "DisjointPathsPlusOne G v0 v1 paths P_new"   and "ProofStepInduct_axioms paths sep_size" shows "ProofStepInduct G v0 v1 paths P_new sep_size"
2: fixes P :: "nat \<Rightarrow> 'a stream \<Rightarrow> bool"   and k :: "nat"   and w :: "'a stream" assumes "P (Suc k) w" shows "\<exists>kaa waa. sdrop k (stl w) = sdrop kaa waa \<and> P kaa waa"
conversion_refl: fixes a :: "'b"   and A :: "('b \<times> 'b) set" shows "(a, a) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
nnf_simps_3: fixes P :: "bool"   and Q :: "bool" shows "(P \<longrightarrow> Q) = (\<not> P \<or> Q)"
sum_multiset_singleton: fixes A :: "'a set" shows "(\<Sum>n\<in>A. {#n#}) = mset_set A"
assms_3: shows "free_vars e1 \<subseteq> V"
ndepth_Push_Node_aux: fixes i :: "nat"   and f :: "nat \<Rightarrow> 'a + nat"   and k :: "nat" shows "(case k of 0 \<Rightarrow> Inr (Suc i) | Suc x \<Rightarrow> f x) = Inr 0 \<longrightarrow> Suc (LEAST x. f x = Inr 0) \<le> k"
natUnion_swap: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> 'b set" shows "natUnion (\<lambda>i. natUnion (f i)) = natUnion (\<lambda>j. natUnion (\<lambda>i. f i j))"
not\<pi>l: shows "\<forall>i\<le>n''. \<pi>'' i \<noteq> \<pi> l"
subgroup_is_submonoid: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G" shows "submonoid H G"
wide_subdg_obj_eq: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>" shows "\<BB>\<lparr>Obj\<rparr> = \<CC>\<lparr>Obj\<rparr>"
infer_simp_1: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option" shows "infer \<Gamma> Unit = Some TUnit"
v: shows "v \<in> {column (LEAST n. A $ i $ n \<noteq> (0::'a)) A |i. row i A \<noteq> 0}"
wfp_on_imp_minimal: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "wfp_on P A" shows "\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q)"
keys_monom_multI: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and v :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'e"   and c :: "'e"   and t :: "'c" assumes "term_powerprod pair_of_term term_of_pair"   and "v \<in> keys p"   and "c \<noteq> (0::'e)" shows "term_powerprod.splus pair_of_term term_of_pair t v \<in> keys (term_powerprod.monom_mult pair_of_term term_of_pair c t p)"
infsetsum_Sigma: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and f :: "'d \<times> 'e \<Rightarrow> 'f" assumes "countable A"   and "\<And>i. countable (B i)"   and "f abs_summable_on Sigma A B" shows "infsetsum f (Sigma A B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B x. f (x, y))"
cases: fixes a1 :: "('fun, 'var) strand_step list set \<times> ('fun, 'var) extstrand_step list"   and a2 :: "('fun, 'var) strand_step list set \<times> ('fun, 'var) extstrand_step list" assumes "a1 \<Rightarrow>\<^sup>\<bullet>\<^sub>c a2" obtains \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) extstrand_step list" where "a1 = (\<S>''__, \<A>''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>''__ [], \<A>''__)" and "[] \<in> \<S>''__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) extstrand_step list" where "a1 = (\<S>'''__, \<A>'''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>'''__ (send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__), \<A>'''__ @ [Step (receive\<langle>t__\<rangle>\<^sub>s\<^sub>t)])" and "send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__ \<in> \<S>'''__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) extstrand_step list" where "a1 = (\<S>''''__, \<A>''''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>''''__ (receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__), \<A>''''__ @ [Step (send\<langle>ta__\<rangle>\<^sub>s\<^sub>t)])" and "receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__ \<in> \<S>''''__"   | a :: "poscheckvariant" and t :: "('fun, 'var) Term.term" and t' :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) extstrand_step list" where "a1 = (\<S>'''''__, \<A>'''''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>'''''__ (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__), \<A>'''''__ @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])" and "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__ \<in> \<S>'''''__"   | X :: "'var list" and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) extstrand_step list" where "a1 = (\<S>''''''__, \<A>''''''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>''''''__ (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S), \<A>''''''__ @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>''''''__"   | f :: "'fun" and T :: "('fun, 'var) Term.term list" and \<A> :: "('fun, 'var) extstrand_step list" and \<S> :: "('fun, 'var) strand_step list set" where "a1 = (\<S>, \<A>)" and "a2 = (\<S>, \<A> @ [Decomp (Fun f T)])" and "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t \<A> \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>"
one_dimvec_def: shows "1 = dim_lambda (\<lambda>i. 0::'c)"
poly_deg_subst_pp_eq_zeroI: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>x. x \<in> keys t \<Longrightarrow> poly_deg (f x) = 0" shows "poly_deg (subst_pp f t) = 0"
sum_root_unity_power_pos_implies_1: fixes b :: "'a \<Rightarrow> complex"   and x :: "'a \<Rightarrow> complex"   and I :: "'a set" assumes "\<And>k. 0 < Re (\<Sum>i\<in>I. b i * x i ^ k)"   and "\<And>i. i \<in> I \<Longrightarrow> \<exists>d. d \<noteq> 0 \<and> x i ^ d = 1" shows "1 \<in> x ` I"
pCons_eq_iff: fixes a :: "'c"   and p :: "'c poly"   and b :: "'c"   and q :: "'c poly" shows "(pCons a p = pCons b q) = (a = b \<and> p = q)"
dom_augment: fixes m :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and S :: "nat set" shows "dom (m \<oplus>\<^bsub>W\<^esub> S) = dom m \<union> S"
inner_eq_zero_iff: fixes x :: "('a, 'cols) vec" shows "(inner_cols x x = (0::'a)) = (x = 0)"
cont_tf_thm_3: fixes v :: "i" shows "[\<^bold>\<exists>p. ContingentlyTrue p in v]"
fold_map_idx_fst_snd_eq: fixes f :: "nat \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> bool \<times> 'a"   and i :: "nat"   and c :: "bool"   and xs :: "'a list" assumes "\<And>i c x. fst (f i c x) = (c \<or> x \<noteq> snd (f i c x))" shows "fst (fold_map_idx f i c xs) = (c \<or> xs \<noteq> snd (fold_map_idx f i c xs))"