hcju/isabelleps huggingface dataset:
simps_1: fixes p :: "'t \<Rightarrow>\<^sub>0 'b" shows "new_spairs [] p = []"
Re_divide_Reals: fixes r :: "complex"   and z :: "complex" assumes "r \<in> \<real>" shows "Re (z / r) = Re z / Re r"
frontier_not_empty: fixes S :: "'b set" assumes "S \<noteq> {}"   and "S \<noteq> UNIV" shows "frontier S \<noteq> {}"
dependent_insertD: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and S :: "'e set" assumes "vector_space scale"   and "a \<notin> module.span scale S"   and "module.dependent scale (insert a S)" shows "module.dependent scale S"
hyp: shows "LeftDerivationFix a i D j b = (is_sentence a \<and> is_sentence b \<and> LeftDerivation a D b \<and> i < length a \<and> j < length b \<and> a ! i = b ! j \<and> (\<exists>E F. D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take i a) E (take j b) \<and> LeftDerivation (drop (Suc i) a) F (drop (Suc j) b)))"
degree_filter_pdevs_le: fixes I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a pdevs" shows "degree (filter_pdevs I x) \<le> degree x"
that: fixes Hs :: "'a set list" assumes "\<And>H. H \<in> set Hs \<Longrightarrow> subgroup H G"   and "cycle Hs"   and "DirProds (\<lambda>H. G\<lparr>carrier := H\<rparr>) (set Hs) \<cong> G"   and "successively (dvd) (map card Hs)"   and "card (set Hs) \<le> card gen"   and "compl_fam id (set Hs)"   and "{\<one>} \<notin> set Hs" shows "thesis"
no_\<tau>move1_\<tau>s_to_no_\<tau>move2: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2"   and "\<And>s1'. \<not> s1 -\<tau>1\<rightarrow> s1'" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> (\<forall>s2''. \<not> s2' -\<tau>2\<rightarrow> s2'') \<and> s1 \<approx> s2'"
invar4_def: shows "invar4 \<equiv> \<lambda>(c, n, ts). c < N \<and> (\<forall>t<N. \<forall>tk. has_ticket (ts t) tk \<longrightarrow> tk < N)"
unique_connecting_path: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "\<exists>!xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
simps: fixes k :: "nat" shows "exexH_k k = tensor_P (exH_k k) (1\<^sub>m K)"
adjugate_def: fixes A :: "('c, 'd) sq_matrix" shows "Square_Matrix.adjugate A = Square_Matrix.transpose (Square_Matrix.cofactor A)"
seE_fringe3: fixes re :: "('d \<times> nat) edge"   and prb :: "('d, 'e, 'f) pre_RedBlack"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and c' :: "('e, 'f) conf" assumes "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"   and "ui_edge ` out_edges (red prb') (src re) \<subset> out_edges (black prb) (fst (src re))"   and "out_edges (black prb) (fst (tgt re)) = {}" shows "fringe prb' = fringe prb"
DmdStable: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> stable P \<and> \<diamond>P \<longrightarrow> \<diamond>\<box>P"
sign_is_surj: fixes n :: "nat" assumes "2 \<le> n" shows "sign ` carrier (sym_group n) = carrier sign_img"
sym_trancl: fixes r :: "('a \<times> 'a) set" assumes "sym r" shows "sym (r\<^sup>+)"
l_new_element_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<equiv> l_new_element type_wf \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_new_element_get_shadow_root_axioms get_shadow_root get_shadow_root_locs"
grid_grid_inj_on: fixes ds :: "nat set"   and ds' :: "nat set"   and b :: "(nat \<times> int) list" assumes "ds \<inter> ds' = {}" shows "inj_on snd (\<Union>p'\<in>grid b ds. \<Union>p''\<in>grid p' ds'. {(p', p'')})"
assms_2: shows "finite {v. f2 v \<noteq> 0}"
abelian_relative_homology_group: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "comm_group (relative_homology_group p X S)"
iso_def: fixes f :: "'c" shows "C.iso f \<equiv> \<exists>g. C.inverse_arrows f g"
notI: fixes P :: "bool" assumes "P \<Longrightarrow> False" shows "\<not> P"
finite_dom_filter_tab: fixes t :: "'a \<Rightarrow> 'b option"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite (dom t)" shows "finite (dom (filter_tab C t))"
major: shows "(red G)\<^sup>*\<^sup>* p q"
jcons_def: shows "jcons \<equiv> all xx (neg (cnj (PP (Var xx)) (PP (encF N (Var xx)))))"
set_prime_factors_nat: fixes n :: "nat" shows "set (prime_factors_nat n) = prime_factors n"
funionM_bind: fixes m :: "'m"   and f :: "'a fset \<Rightarrow> 'm"   and g :: "'a fset \<Rightarrow> 'm" shows "funionM (bind m f) (bind m g) = bind m (\<lambda>x. funionM (f x) (g x))"
value_to_sterm: fixes v :: "value" assumes "vconstructor_value v" shows "is_value (value_to_sterm v)"
pwd_spy_dec_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_dec n)"
prevVertex_in_face: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "f\<^bsup>-1\<^esup> \<bullet> v \<in> \<V> f"
left_nesting_simps_3: fixes t :: "'c"   and u :: "'c" shows "left_nesting (app t u) = Suc (left_nesting t)"
plus_one_helper2: fixes x :: "'b word"   and n :: "'b word" assumes "x \<le> n"   and "n + 1 \<noteq> 0" shows "x < n + 1"
bal_add0: fixes t :: "'a tree23"   and k :: "int"   and y :: "'a" assumes "bal t" shows "bal (add0 k y t)"
distinct_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist"   and x1 :: "'b" shows "TCons x21 x22 \<noteq> TNil x1"
not_in_Gromov_boundary': fixes x :: "'a" shows "to_Gromov_completion x \<notin> Gromov_boundary"
unitr_eq: fixes e :: "'a"   and x :: "'a"   and y :: "'a" assumes "e \<in> \<xi>"   and "\<rho> x y e" shows "x = y"
streams_empty_iff: fixes S :: "'b set" shows "(streams S = {}) = (S = {})"
known_ptr_def: shows "DocumentClass.known_ptr \<equiv> l_known_ptr\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t.a_known_ptr"
lit_one: shows "\<guillemotleft>1::'c\<guillemotright> = 1"
decompose_Some: fixes f :: "'a"   and ss :: "('a, 'b) Term.term list"   and g :: "'a"   and ts :: "('a, 'c) Term.term list"   and E :: "(('a, 'b) Term.term \<times> ('a, 'c) Term.term) list" assumes "decompose (Fun f ss) (Fun g ts) = Some E" shows "f = g \<and> length ss = length ts \<and> E = zip ss ts"
ab_nz: shows "a \<noteq> 0 \<or> b \<noteq> 0"
M1_mult_ket_k: fixes k :: "nat" assumes "k < R" shows "M1 *\<^sub>v ket_k k = ket_k k"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_f32]"
OrderingSetIso_pow_complement: fixes A :: "'b set" shows "OrderingSetIso order.greater_eq order.greater dual_order.greater_eq dual_order.greater (Pow A) ((-) A)"
simps_7: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
simps: fixes v :: "nat \<Rightarrow> 'a"   and lb :: "nat \<Rightarrow> 'a option"   and ub :: "nat \<Rightarrow> 'a option"   and S :: "nat set" shows "v \<Turnstile>\<^sub>b (lb, ub) \<parallel> S = (\<forall>x\<in>S. in_bounds x v (lb, ub))"
rcos_ring_hom_ring: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R" shows "ring_hom_ring R (R Quot I) ((+>\<^bsub>R\<^esub>) I)"
preserves_cod: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "partial_magma.cod B (\<tau> f) = G (partial_magma.cod A f)"
amtx_cnv: fixes M :: "nat" shows "amtx_assn N M id_assn = IICF_Array_Matrix.is_amtx N M"
succs_1: shows "succs i \<tau> pc \<noteq> []"
simps: fixes xs :: "('b, 'a) tllist" shows "terminal0 xs = (if is_TNil xs then case xs of TNil x \<Rightarrow> id x | TCons x xa \<Rightarrow> undefined x xa else terminal0 (ttl xs))"
simps_5: fixes v :: "nat" shows "eta_norm (Bv v) = Bv v"
IS_RIGHT_UNIQUED: fixes r :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
row_echelon_form_dim0_col: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n 0" shows "row_echelon_form A"
is_hsum_def: fixes z :: "hf" shows "is_hsum z = (\<exists>x. z = HF.Inl x \<or> z = HF.Inr x)"
nn_integral_monotone_convergence_SUP_countable: fixes Y :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and B :: "'d set" assumes "Y \<noteq> {}"   and "Complete_Partial_Order.chain (\<le>) (f ` Y)"   and "countable B" shows "\<integral>\<^sup>+ x. (\<Squnion>i\<in>Y. f i x) \<partial>count_space B = (\<Squnion>i\<in>Y. integral\<^sup>N (count_space B) (f i))"
gm_real: fixes x :: "'a" shows "enn2real (gm x) = max (- g x) 0"
simps_1: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op" shows "Smith_Impl.Smith_1xn_aux Smith_1x2 0 A (S, Q) = (S, Q)"
weak_composition_with_homs_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "weak_composition_with_homs V H src trg \<equiv> weak_composition V H \<and> horizontal_homs V src trg \<and> weak_composition_with_homs_axioms V H src trg"
of_weak_ranking_Cons: fixes z :: "'b set"   and zs :: "'b set list"   and y :: "'b"   and x :: "'b" shows "of_weak_ranking (z # zs) y x = (x \<in> z \<and> y \<in> \<Union> (set (z # zs)) \<or> of_weak_ranking zs y x)"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "C.par f f'"   and "C.L f = C.L f'" shows "f = f'"
\<psi>_mapsto: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and b :: "'a"   and a :: "'a" assumes "hom_functor C S \<phi>"   and "partial_magma.ide C b"   and "partial_magma.ide C a" shows "hom_functor.\<psi> C \<phi> (b, a) \<in> hom_functor.set C \<phi> (b, a) \<rightarrow> {f. partial_magma.in_hom C f b a}"
SubstFormP_Disj: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstFormP v i x x', SubstFormP v i y y'} \<turnstile> SubstFormP v i (Q_Disj x y) (Q_Disj x' y')"
shadows_consts_def: fixes t :: "'a" shows "shadows_consts t = (\<not> fdisjnt all_consts (all_frees t))"
A: shows "atoms_pltl \<phi> \<subseteq> A"
init_def: shows "init = (card (vertices (fst R)), edges (fst R))"
ord_le: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G"   and "x [^] n = \<one>"   and "0 < n" shows "ord x \<le> n"
completeE: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open"   and "uniform_space.complete open uniformity s"   and "\<forall>n. f n \<in> s"   and "uniform_space.Cauchy uniformity f" obtains l :: "'b" where "l \<in> s" and "filterlim f (topological_space.nhds open l) sequentially"
AbsS: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>A!,x\<rparr> in v] = (\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<alpha>\<nu> y)"
pochhammer_absorb_comp: fixes r :: "'a"   and k :: "nat" shows "(r - of_nat k) * pochhammer (- r) k = r * pochhammer (- r + (1::'a)) k"
r_smn_prim: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "prim_recfn (Suc m) (r_smn n m)"
RDirProd_list_hom1: fixes R :: "('a, 'b) ring_scheme"   and Rs :: "('a, 'b) ring_scheme list" shows "(\<lambda>(a, as). a # as) \<in> ring_hom (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))"
invariant_iff_pos_and_neg_invariant: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.invariant f X M = (auto_ll_on_open.positively_invariant f X M \<and> auto_ll_on_open.negatively_invariant f X M)"
n_mult_def: fixes f :: "'a \<Rightarrow> nat \<Rightarrow> int" shows "n_mult f = (\<lambda>n. [n] \<cdot> f n)"
CompDefOp: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "f \<approx>>\<^bsub>C\<^esub> g" shows "g \<approx>>\<^bsub>Op C\<^esub> f"
block_exists: fixes a :: "'a" assumes "a \<in> Ker" shows "\<exists>A. a \<in> A \<and> A \<in> L.Partition"
smult_vec_length_bis: fixes x :: "real"   and v :: "complex vec" assumes "0 \<le> x" shows "\<parallel>col (complex_of_real x \<cdot>\<^sub>m |v\<rangle>) 0\<parallel> = x * \<parallel>v\<parallel>"
approximating_bigstep_fun_seq_Undecided_t_wf: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list"   and t :: "state" assumes "wf_ruleset \<gamma> p (rs1 @ rs2)" shows "(approximating_bigstep_fun \<gamma> p (rs1 @ rs2) Undecided = t) = (approximating_bigstep_fun \<gamma> p rs1 Undecided = Undecided \<and> approximating_bigstep_fun \<gamma> p rs2 Undecided = t \<or> approximating_bigstep_fun \<gamma> p rs1 Undecided = t \<and> t \<noteq> Undecided)"
is_unitE: fixes a :: "'a"   and c :: "'a" assumes "is_unit a" obtains b :: "'a" where "a \<noteq> (0::'a)" and "b \<noteq> (0::'a)" and "is_unit b" and "(1::'a) div a = b" and "(1::'a) div b = a" and "a * b = (1::'a)" and "c div a = c * b"
gv_3: fixes x :: "'a" shows "gv x < \<infinity>"
normalize_mult_unit_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times b one" shows "normalize (times a b) = normalize a"
map_2: shows "G\<^sub>3 = map_ugraph f\<^sub>2 G\<^sub>2"
induct: fixes P :: "'a acom \<Rightarrow> 'a acom \<Rightarrow> bool"   and a0 :: "'a acom"   and a1 :: "'a acom" assumes "\<And>S S'. P (SKIP {S}) (SKIP {S'})"   and "\<And>x e S x' e' S'. P (x ::= e {S}) (x' ::= e' {S'})"   and "\<And>c1 c2 c1' c2'. \<lbrakk>P c1 c1'; P c2 c2'\<rbrakk> \<Longrightarrow> P (c1;; c2) (c1';; c2')"   and "\<And>b c1 c2 S b' c1' c2' S'. \<lbrakk>P c1 c1'; P c2 c2'\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2 {S}) (IF b' THEN c1' ELSE c2' {S'})"   and "\<And>Inv b c Pa Inv' b' c' P'. P c c' \<Longrightarrow> P ({Inv} WHILE b DO c {Pa}) ({Inv'} WHILE b' DO c' {P'})"   and "\<And>v va vb vc. P (v ::= va {vb}) (SKIP {vc})"   and "\<And>v va vb vc vd. P (v ::= va {vb}) (vc;; vd)"   and "\<And>v va vb vc vd ve vf. P (v ::= va {vb}) (IF vc THEN vd ELSE ve {vf})"   and "\<And>v va vb vc vd ve vf. P (v ::= va {vb}) ({vc} WHILE vd DO ve {vf})"   and "\<And>v va vb. P (v;; va) (SKIP {vb})"   and "\<And>v va vb vc vd. P (v;; va) (vb ::= vc {vd})"   and "\<And>v va vb vc vd ve. P (v;; va) (IF vb THEN vc ELSE vd {ve})"   and "\<And>v va vb vc vd ve. P (v;; va) ({vb} WHILE vc DO vd {ve})"   and "\<And>v va vb vc vd. P (IF v THEN va ELSE vb {vc}) (SKIP {vd})"   and "\<And>v va vb vc vd ve vf. P (IF v THEN va ELSE vb {vc}) (vd ::= ve {vf})"   and "\<And>v va vb vc vd ve. P (IF v THEN va ELSE vb {vc}) (vd;; ve)"   and "\<And>v va vb vc vd ve vf vg. P (IF v THEN va ELSE vb {vc}) ({vd} WHILE ve DO vf {vg})"   and "\<And>v va vb vc vd. P ({v} WHILE va DO vb {vc}) (SKIP {vd})"   and "\<And>v va vb vc vd ve vf. P ({v} WHILE va DO vb {vc}) (vd ::= ve {vf})"   and "\<And>v va vb vc vd ve. P ({v} WHILE va DO vb {vc}) (vd;; ve)"   and "\<And>v va vb vc vd ve vf vg. P ({v} WHILE va DO vb {vc}) (IF vd THEN ve ELSE vf {vg})"   and "\<And>vc v va vb. P (SKIP {vc}) (v ::= va {vb})"   and "\<And>vc vd v va vb. P (vc;; vd) (v ::= va {vb})"   and "\<And>vc vd ve vf v va vb. P (IF vc THEN vd ELSE ve {vf}) (v ::= va {vb})"   and "\<And>vc vd ve vf v va vb. P ({vc} WHILE vd DO ve {vf}) (v ::= va {vb})"   and "\<And>vb v va. P (SKIP {vb}) (v;; va)"   and "\<And>vb vc vd v va. P (vb ::= vc {vd}) (v;; va)"   and "\<And>vb vc vd ve v va. P (IF vb THEN vc ELSE vd {ve}) (v;; va)"   and "\<And>vb vc vd ve v va. P ({vb} WHILE vc DO vd {ve}) (v;; va)"   and "\<And>vd v va vb vc. P (SKIP {vd}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve vf v va vb vc. P (vd ::= ve {vf}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve v va vb vc. P (vd;; ve) (IF v THEN va ELSE vb {vc})"   and "\<And>vd ve vf vg v va vb vc. P ({vd} WHILE ve DO vf {vg}) (IF v THEN va ELSE vb {vc})"   and "\<And>vd v va vb vc. P (SKIP {vd}) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve vf v va vb vc. P (vd ::= ve {vf}) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve v va vb vc. P (vd;; ve) ({v} WHILE va DO vb {vc})"   and "\<And>vd ve vf vg v va vb vc. P (IF vd THEN ve ELSE vf {vg}) ({v} WHILE va DO vb {vc})" shows "P a0 a1"
irreducible\<^sub>d_factorization_exists_1: fixes p :: "'a poly" assumes "0 < degree p" shows "\<exists>fs. fs \<noteq> [] \<and> (\<forall>f\<in>set fs. irreducible\<^sub>d f \<and> degree f \<le> degree p) \<and> p = prod_list fs"
inject_5: fixes x6 :: "'e"   and y6 :: "'e" shows "(Inca.instr.IOpInl x6 = Inca.instr.IOpInl y6) = (x6 = y6)"
closed_shiftpath: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "pathfinish (shiftpath a g) = pathstart (shiftpath a g)"
inf_bot_right: fixes x :: "'a multiset" shows "x \<inter># {#} = {#}"
maximal_drop: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist"   and n :: "nat" assumes "Digraph G"   and "Digraph.maximal_path G P" shows "Digraph.maximal_path G (ldropn n P)"
RDirProd_list_def: fixes Rs :: "('a, 'b) ring_scheme list" shows "RDirProd_list Rs = foldr (\<lambda>R S. image_ring (\<lambda>(a, as). a # as) (RDirProd R S)) Rs nil_ring"
osum_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "trans r"   and "trans r'" shows "trans (r +o r')"
remainder_recovery_bis: fixes K :: "'a set"   and A :: "'a set"   and \<psi> :: "'a"   and B :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K \<turnstile> \<psi>"   and "\<not> B \<turnstile> \<psi>"   and "B \<in> K .\<bottom>. \<phi>" shows "B \<in> K .\<bottom>. \<psi>"
inner_vec_def: fixes x :: "('a, 'n) vec"   and y :: "('a, 'n) vec" shows "inner x y = (\<Sum>i\<in>UNIV. inner_field (x $ i) (y $ i))"
individually_rational_on_Cd: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "PdXXX.individually_rational_on ds X" shows "PdXXX.Cd d X = PdXXX.dX X d"
iT_iff_3: fixes x :: "nat"   and n :: "nat"   and d :: "nat" shows "(x \<in> [n\<dots>,d]) = (n \<le> x \<and> x \<le> n + d)"
termdiffs: fixes c :: "nat \<Rightarrow> 'a"   and K :: "'a"   and x :: "'a" assumes "summable (\<lambda>n. c n * K ^ n)"   and "summable (\<lambda>n. diffs c n * K ^ n)"   and "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"   and "norm x < norm K" shows "((\<lambda>x. \<Sum>n. c n * x ^ n) has_field_derivative (\<Sum>n. diffs c n * x ^ n)) (at x)"
small_big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> l F g"   and "g \<in> L F h" shows "f \<in> l F h"
strict_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "strict_suffix a b"   and "strict_suffix b c" shows "strict_suffix a c"
general_preserved: fixes T :: "ty"   and S :: "tyS"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "T \<prec> S" shows "\<theta><T> \<prec> \<theta><S>"
that: fixes p :: "'a list"   and y :: "'a" assumes "path G x p y"   and "y \<in> B"   and "\<And>z. z \<in> set (x # p) \<Longrightarrow> z \<notin> S" shows "thesis"
simps_1: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.digitToList (One a) = FingerTreeStruc.nodeToList a"
LI_converse_wellfounded_trans: shows "wf ((LI_rel\<^sup>+)\<inverse>)"
comp_of_ord_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "comp_of_ord le x y = (if le x y then if x = y then Eq else Lt else Gt)"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('g, 'h) \<I>"   and gpv :: "('i, 'g, 'h) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
I'_carr: shows "I \<in> Suc ` {..n} \<rightarrow> carrier (ideals_set R)"
KeepProperty: fixes low :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "\<forall>i\<ge>low. P i \<longrightarrow> P (Suc i) \<and> Q i"   and "P low" shows "\<forall>i\<ge>low. Q i"
arc_balancedI_trail: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.trail G u p v" shows "pre_digraph.arc_set_balanced G u (set p) v"
conn1A: fixes k :: "cnf" shows "bbrp (\<sigma>\<^bsub>the_bb\<^esub>k) = (\<Union>sid\<in>{sid. \<parallel>sid\<parallel>\<^bsub>k\<^esub>}. {ksrp (\<sigma>\<^bsub>sid\<^esub>k)})"
ideal_generated_eq_left_ideal: 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 A :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "ring.ideal_generated plus zero uminus times A = ring.left_ideal_generated plus zero uminus times A"
face_distance_le: fixes C :: "'a set"   and F :: "'a set"   and D :: "'a set" assumes "chamber C"   and "order.greater_eq C F" shows "order.greater_eq (chamber_distance C D) (face_distance F D)"
cut_le_mono: fixes t :: "'a"   and t' :: "'a"   and I :: "'a set" assumes "t \<le> t'" shows "I \<down>\<le> t \<subseteq> I \<down>\<le> t'"
proj_stoch_proc_def: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and n :: "nat" shows "proj_stoch_proc f n = (\<lambda>w. smap_stoch_proc n f 0 w @- sconst (f n w))"
composite_functor_axioms: shows "composite_functor C.CC.comp (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) T\<^sub>C F"
otherassms_5: shows "(\<xi>, pp) \<in> reachable A I"
BisT_sym: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "sym (PL_Indis.BisT tval aval indis)"
cut: fixes G' :: "'a triple set"   and ts :: "'a triple set"   and G :: "'a triple set" assumes "G'||-ts"   and "G||-G'" shows "G||-ts"