hcju/isabelleps huggingface dataset:
valid_line_def: fixes l :: "color list" shows "valid_line l = (\<forall>i<length l. valid_point l i)"
gfp_trans_unitary: fixes P :: "'a \<Rightarrow> real"   and T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "unitary P" shows "unitary (gfp_trans T P)"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev GM a \<noteq> a"
inf_last_ti_Suc2: fixes dt :: "nat \<Rightarrow> 'a list"   and t :: "nat"   and k :: "nat" assumes "dt (Suc t) \<noteq> [] \<or> dt (Suc (Suc t)) \<noteq> []" shows "inf_last_ti dt (t + 2 + k) \<noteq> []"
unsigned_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero 1 = one"
smult_as_map_poly: fixes a :: "'a" shows "Polynomial.smult a = map_poly ((*) a)"
interval_mul_commute: fixes A :: "'b interval"   and B :: "'b interval" shows "A * B = B * A"
nths_split_complement_eq: fixes A :: "nat set"   and B :: "nat set"   and xs :: "'a list" assumes "A \<inter> B = {}"   and "{0..<length xs} \<subseteq> A \<union> B" shows "nths xs A = nths xs (- B)"
hyp: shows "span (cols_upt_k A k) = span (cols_upt_k (Gram_Schmidt_column_k A j) k)"
update_conv_update_with: fixes k :: "'a"   and v :: "'b" shows "DAList.update k v = update_with v k (\<lambda>_. v)"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (CCC.prod f b)"
HEndPhase0_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "HEndPhase0 s s' p = (EndPhase0 s s' p \<and> HNextPart s s')"
rref_drop_pivots: fixes R :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "row_echelon_form R"   and "R \<in> carrier_mat n nc"   and "n \<le> nc" shows "vec_space.rank n (take_cols R (map snd (pivot_positions R))) = vec_space.rank n R"
disc_put_option_lgeom: fixes T :: "nat"   and K :: "real"   and w :: "bool stream" shows "discounted_value r (\<lambda>m. put_option T K) T w = det_discount r T * max (K - lgeom_proc u d init (stake T w)) 0"
nn_integral_T: fixes f :: "'a cfg stream \<Rightarrow> ennreal"   and s :: "'a cfg" assumes "f \<in> borel_measurable (stream_space (count_space UNIV))" shows "integral\<^sup>N (MC.T s) f = \<integral>\<^sup>+ t. \<integral>\<^sup>+ \<omega>. f (t ## \<omega>) \<partial>MC.T t \<partial>measure_pmf (K_cfg s)"
wp1_auxa: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "wellformed_policy1_strong p" shows "\<exists>r. applied_rule_rev Cp x p = \<lfloor>r\<rfloor>"
wn0: fixes w :: "complex" assumes "w \<notin> S" shows "winding_number p w = 0"
IH: assumes "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel S)" shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t S))"
intros: fixes classes :: "'a fset"   and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and \<C> :: "'a"   and "from" :: "'c option"   and role :: "'c"   and \<D>' :: "'a"   and end' :: "'a \<times> nat \<times> enat \<times> bool \<times> bool"   and \<D> :: "'a"   and "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "closest_association_end classes associations \<C> from role \<D>' end'"   and "\<D> \<noteq> \<D>' \<or> end \<noteq> end'" shows "closest_association_end_not_unique classes associations \<C> from role \<D> end"
welltyped_Abs: fixes T :: "Unary_PCF.type"   and X :: "(char list \<times> Unary_PCF.type) fset"   and e :: "expr"   and U :: "Unary_PCF.type" assumes "\<forall>x. (x, T) |\<notin>| X \<longrightarrow> open0_Var (x, T) e ::: U" shows "\<Lambda>\<langle>T\<rangle> e ::: T \<rightarrow> U"
representation_ne_zero: fixes basis :: "'d set"   and v :: "'d"   and b :: "'d" assumes "crepresentation basis v b \<noteq> 0" shows "b \<in> basis"
T1: shows "S.UP ((g \<star> \<epsilon>) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g)) = S.cmp\<^sub>U\<^sub>P (g, src g) \<cdot>\<^sub>S (S.UP g \<star>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g)) \<cdot>\<^sub>S (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<star>\<^sub>S S.UP g) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (trg g, g))"
xor_int_rec: fixes k :: "int"   and l :: "int" shows "k XOR l = of_bool (odd k \<noteq> odd l) + 2 * (k div 2 XOR l div 2)"
sdom_dom_succs: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "strict_dom i j"   and "j \<in> succs k" shows "local.dom i k"
aezfun_setspan_proj_zero: fixes A :: "'b set" shows "aezfun_setspan_proj A 0 = 0"
fresh_7: fixes a :: "name"   and x2 :: "coname"   and x3 :: "trm"   and x1 :: "name" shows "a \<sharp> NotL <x2>.x3 x1 = (a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
mod_pure_star_dist: fixes P :: "assn"   and b :: "bool"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P * \<up> b) = (h \<Turnstile> P \<and> b)"
aodv_init_sip_not_ip': fixes \<xi> :: "state"   and p :: "(state, msg, pseqp, pseqp label) seqp"   and i :: "nat" assumes "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i" shows "sip \<xi> \<noteq> ip \<xi>"
ascP_Cons: fixes x :: "'d" shows "ascP ((#) x)"
getOneIp_elem: fixes W :: "'a wordinterval" assumes "\<not> wordinterval_empty W" shows "wordinterval_element (getOneIp W) W"
enum_H2_def: shows "enum_class.enum = [H1, H2]"
assms_9: shows "abs_conv_abscissa (fds_deriv f / f) \<le> s0"
soundI: fixes P :: "'c \<Rightarrow> real" assumes "Expectations.bounded P"   and "nneg P" shows "sound P"
ts__suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B TS C D" shows "C B A A B D SumA C B D"
nb\<^sub>2: fixes op :: "'a strips_operator" assumes "op \<in> set \<pi>" shows "are_operator_effects_consistent op op"
symvar_def: fixes v :: "'c"   and s :: "'c \<Rightarrow> nat" shows "symvar v s \<equiv> (v, s v)"
division_segment_not_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "division_segment a \<noteq> zero"
expand: fixes prod :: "'e \<times> 'f"   and prod' :: "'e \<times> 'f" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
y: fixes x :: "'a" assumes "x \<in> closure X" shows "(f \<longlongrightarrow> y x) (at x within X)"
AnalLet_subst: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp"   and a :: "'a" assumes "x \<notin> domA \<Gamma>"   and "y \<notin> domA \<Gamma>" shows "AnalLet \<Gamma>[x::h=y] e[x::=y] a = AnalLet \<Gamma> e a"
rank_eq_dim_col_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = vec.dim (col_space A)"
IH: fixes x :: "'a" assumes "y \<noteq> []"   and "x \<in> set y" shows "index_of x y < length y"
Trg_Src: fixes t :: "'c local.term" assumes "Arr t" shows "Trg (Src t) = Src t"
insert_Collect: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a::'b) \<triangleright> Collect (P::'b \<Rightarrow> bool) = {u::'b. u \<noteq> a \<longrightarrow> P u}"
size_N': fixes \<psi> :: "'a ltln" assumes "\<psi> \<in> N" shows "size (\<psi>[M]\<^sub>\<Pi>\<^sub>1) \<le> size \<phi>"
Conc_not_MT: fixes p :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "p = x # xs" shows "p \<noteq> []"
sw_compose: fixes xs' :: "'c"   and x' :: "'b"   and y' :: "'b"   and zs :: "'c"   and xs :: "'c"   and x :: "'b"   and y :: "'b"   and z :: "'b" shows "z @zs[x \<and> y]_xs @zs[x' \<and> y']_xs' = z @zs[x' \<and> y']_xs' @zs[x @xs[x' \<and> y']_xs' \<and> y @xs[x' \<and> y']_xs']_xs"
step_op_inl_miss: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and opinl :: "'opinl"   and ar :: "nat"   and \<Sigma> :: "'dyn list"   and x :: "'dyn"   and F' :: "'fenv_inca"   and H :: "'henv"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IOpInl opinl"   and "\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl) = ar"   and "ar \<le> length \<Sigma>"   and "\<not> \<II>\<ss>\<II>\<nn>\<ll> opinl (take ar \<Sigma>)"   and "\<II>\<nn>\<ll>\<OO>\<pp> opinl (take ar \<Sigma>) = x"   and "Finca_add F f (rewrite_fundef_body fd pc (Inca.instr.IOp (\<DD>\<ee>\<II>\<nn>\<ll> opinl))) = F'" shows "Sinca.step (State F H (Frame f pc \<Sigma> # st)) (State F' H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
pseudo_mod_2: fixes g :: "'b poly"   and f :: "'b poly" assumes "g \<noteq> 0" shows "pseudo_mod f g = 0 \<or> degree (pseudo_mod f g) < degree g"
that: fixes lb :: "real" assumes "lb < ub"   and "\<forall>z. lb \<le> z \<and> z < ub \<longrightarrow> poly p z \<noteq> 0" shows "thesis"
eventually_const_iff: fixes P :: "bool"   and F :: "'c filter" shows "(\<forall>\<^sub>F x in F. P) = (P \<or> F = \<bottom>)"
zmultiset_nonemptyE: fixes A :: "'a zmultiset" assumes "A \<noteq> {#}\<^sub>z" obtains x :: "'a" where "x \<in>#\<^sub>z A"
typ_ok_def: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and T :: "typ" shows "typ_ok \<Theta> T \<equiv> wf_type (sig \<Theta>) T"
chain_imageI: fixes le_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'c set"   and le_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
\<Z>_csgrp_axioms_def: fixes \<SS> :: "V" shows "\<Z>_csgrp_axioms \<SS> \<equiv> \<forall>a b. a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a"
wellformed_deltas: fixes s :: "'s"   and w :: "'a list" assumes "wellformed s"   and "w \<in> lists (set alphabet)" shows "wellformed (fold delta w s)"
permutes_swap_id: fixes a :: "'b"   and S :: "'b set"   and b :: "'b" assumes "a \<in> S"   and "b \<in> S" shows "Fun.swap a b id permutes S"
collinear_between_cases: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "collinear {a, b, c} = (between (b, c) a \<or> between (c, a) b \<or> between (a, b) c)"
elims: fixes x :: "('a, 'b, 'c) expr"   and xa :: "'c"   and xb :: "('a, 'b, 'c) expr"   and y :: "('a, 'b, 'c) expr" assumes "constant_function x xa xb = y" obtains e :: "('a, 'b, 'c) expr" and x :: "'c" and e' :: "('a, 'b, 'c) expr" where "xa__ = e" and "xa = x" and "xb = e'" and "y = VE (CV Unit)"
zero_is_polynomial: fixes K :: "'a list set" shows "polynomial\<^bsub>K [X]\<^esub> K []"
iTILL_sub2: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "k \<ominus> [\<dots>n] = [\<dots>k]"
P5: shows "\<lfloor>\<lambda>w. \<nexists>x. x actualizedAt w \<and> (\<exists>xa. ((actualizedAt) xa \<^bold>\<and> Contingent xa \<^bold>\<and> Necessary x \<^bold>\<and> xa explains x) w)\<rfloor>"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. fst (f, w) \<star> snd (f, w))"
3: shows "LinearCombinations.module.span K (vs (subspace_sum W1 W2)) (b1 \<union> b2) = carrier (vs (subspace_sum W1 W2))"
sorted_stable_segment: fixes cmp :: "'b comparator"   and x :: "'b"   and xs :: "'b list" shows "Sorting_Algorithms.sorted cmp (filter (\<lambda>y. compare cmp x y = Equiv) xs)"
preserves_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.dom (\<F>C.\<nu> f) = \<F>C.map (\<F>C.dom f)"
IH_1: shows "reduce_nested_mpoly (extract_var p1 v) = p1"
assms_1: shows "\<forall>n. f n \<le> f (Suc n)"
Zp_nat_inc_rep: fixes n :: "nat" shows "[n] \<cdot> \<one> = (\<lambda>m. p_residue m (int n))"
\<rho>_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<rho> a = \<r>[a]"
case_prod_beta': fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'm" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
add_equal_means_equal: fixes \<Gamma> :: "'a multiset"   and A :: "'a"   and \<Delta> :: "'a multiset" assumes "\<Gamma> \<oplus> A = \<Delta> \<oplus> A" shows "\<Gamma> = \<Delta>"
gcd_nat_abs_right_eq: fixes n :: "nat"   and k :: "int" shows "gcd n (nat \<bar>k\<bar>) = nat (gcd (int n) k)"
left_connected: fixes \<nu> :: "'d"   and \<nu>' :: "'d"   and \<mu> :: "'d" assumes "D.seq \<nu> \<nu>'" shows "(\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null) = (\<nu>' \<star>\<^sub>D \<mu> \<noteq> D.null)"
c_K33_not_comb_planar: shows "\<not> comb_planar (with_proj c_K33)"
mult_eq: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "residues_mult_nat n" shows "x \<otimes>\<^bsub>Residues_Mult n\<^esub> y = x * y mod n"
covering_space_lift_strong: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and a :: "'c"   and z :: "'e"   and U :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "covering_space C p S"   and "a \<in> C"   and "z \<in> U"   and "simply_connected U"   and "locally path_connected U"   and "continuous_on U f"   and "f ` U \<subseteq> S"   and "f z = p a" obtains g :: "'e \<Rightarrow> 'c" where "continuous_on U g" and "g ` U \<subseteq> C" and "g z = a" and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"
\<theta>\<^sub>g_def: shows "\<theta>\<^sub>g = \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> fg\<^sub>0h\<^sub>1.\<phi> \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1]"
prob_le_1: fixes s :: "'s"   and A :: "'s stream set" shows "T.prob s A \<le> 1"
frameImpEqvt: fixes p :: "(name \<times> name) list"   and F :: "'b frame"   and \<Phi> :: "'c" shows "p \<bullet> (F \<turnstile>\<^sub>F \<Phi>) = p \<bullet> F \<turnstile>\<^sub>F p \<bullet> \<Phi>"
WT_converterD_results: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and q :: "'a"   and r :: "'b"   and conv' :: "('a, 'b, 'c, 'd) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>"   and "(r, conv') \<in> results_gpv \<I>' (run_converter conv q)" shows "r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>"
nofail_RES_conv: fixes m :: "'b nres" shows "nofail m = (\<exists>M. m = RES M)"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "vec n f \<in> carrier_vec n"
thyps_2: shows "\<forall>\<tau>\<in>{0..t}. \<tau> \<le> - (ln ((L - Tmax) / (L - T)) / a)"
goal3_5: shows "Rmi_g ni2 n2 bdd"
UN_le_eq_Un0_strict: fixes M :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "\<Union> (M ` {..<n + 1}) = \<Union> (M ` {1..<n + 1}) \<union> M 0"
reflect_factorization: fixes p :: "'b poly"   and fctrs :: "'b \<times> ('b \<times> nat) list" assumes "constant_term p \<noteq> (0::'b)"   and "is_factorization_of fctrs p" shows "is_alt_factorization_of fctrs (reflect_poly p)"
Suc: fixes q :: "'a poly"   and r :: "'a poly"   and dr :: "nat"   and q' :: "'a poly" shows "snd (pseudo_divmod_main (x * lc) q r (Polynomial.smult x d) dr n) = snd (pseudo_divmod_main lc q' (Polynomial.smult (x ^ n) r) d dr n)"
Sup_bool_def: fixes A :: "bool set" shows "Sup A = (True \<in> A)"
1: fixes p :: "'a"   and q :: "'a" shows "\<exists>k. p * (a1 * d) + q * (b1 * d) = k * d"
cut_less_not_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<notin> I" shows "x \<notin> I \<down>< t"
set_subset_Univ: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.ide S a" shows "set_category_data.set S img a \<subseteq> Collect (category.terminal S)"
asm: shows "\<forall>x\<in>vars\<^sub>t\<^sub>s ts. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
leg1_composite: shows "doms.\<nu>\<pi>.leg1 = doms.\<nu>.leg1 \<cdot> doms.\<nu>\<pi>.prj\<^sub>1"
distinct_5: fixes x4 :: "ty" shows "NullT \<noteq> ArrayT x4"
oc_projection_unique: fixes i :: "nat"   and v :: "rat vec" assumes "i < length (map of_int_hom.vec_hom fs)"   and "v \<in> Rn"   and "\<And>x. x \<in> gs1.gso ` {0..<i} \<Longrightarrow> v \<bullet> x = 0"   and "map of_int_hom.vec_hom fs ! i - v \<in> gs.span (gs1.gso ` {0..<i})" shows "v = gs1.gso i"
419: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - - (- x \<sqinter> y) = - - (- x \<sqinter> y)"
dom_part: fixes s :: "char list \<Rightarrow> int" shows "dom (part s) = UNIV"
simps_2: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and a :: "('a, 'b) Node"   and v :: "('a, 'b) Node"   and va :: "('a, 'b) Node list" shows "FingerTreeStruc.splitNlist p i (a # v # va) = (let i2 = i + FingerTreeStruc.gmn a in if p i2 then ([], a, v # va) else let (l, x, r) = FingerTreeStruc.splitNlist p i2 (v # va) in (a # l, x, r))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p. (\<And>x. P p) \<Longrightarrow> P (E p)"   and "\<And>p. (\<And>x. P (NOT p)) \<Longrightarrow> P (A p)"   and "\<And>p. (\<And>x. P p) \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (Le v)"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)" shows "P a0"
hom: shows "homeomorphism U' V (blinfun_apply invf \<circ> (\<lambda>x. (f \<circ> (+) x0) x - f x0)) g"
prebicategory_with_units_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b" shows "prebicategory_with_units V H \<a> \<i> \<equiv> prebicategory V H \<a> \<and> weak_composition V H \<and> prebicategory_with_units_axioms V H \<i>"
completenessForSAT: fixes F0 :: "Literal list list"   and state0 :: "State"   and state :: "State"   and decisionVars :: "nat set" assumes "satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "getConflictFlag state = False \<and> \<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state))"
ide_pbdom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.ide C (elementary_category_with_pullbacks.pbdom C prj0 f g)"
scale_left_imp_eq: fixes a :: "'b"   and x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec" assumes "a \<noteq> (0::'b)"   and "a *s x = a *s y" shows "x = y"
inj_on_the_inv_into: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "inj_on (the_inv_into A f) (f ` A)"
simps_1: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and P :: "'b"   and tset :: "('f, 'c, 'd, 'e) Tree set['f]" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P (tConj tset) = Ball (set_bset tset) (effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply P)"
dlex_pm_lin: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "dlex_pm s t \<or> dlex_pm t s"
nGt_le_conv: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and x :: "'c"   and y :: "'c" assumes "comparator comp" shows "(comp x y \<noteq> Gt) = le_of_comp comp x y"
fract_poly_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "map_poly to_fract (p * q) = map_poly to_fract p * map_poly to_fract q"
simps: fixes a1 :: "lam"   and a2 :: "lam" shows " a1 \<longrightarrow>\<^sub>\<beta>\<^sup>* a2 = ((\<exists>t. a1 = t \<and> a2 = t) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and>  t \<longrightarrow>\<^sub>\<beta> s) \<or> (\<exists>t1 t2 t3. a1 = t1 \<and> a2 = t3 \<and>  t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2 \<and>  t2 \<longrightarrow>\<^sub>\<beta>\<^sup>* t3))"
carrier_def: shows "W'.src.carrier = \<Union> (domain ` charts_submanifold W')"
is_generator_set: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and canonical_basis :: "'b list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "cspan (set canonical_basis) = UNIV"
EncI: fixes F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>"   and v :: "i" assumes "\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.en r" shows "[\<lbrace>x,F\<rbrace> in v]"
ext_inject: fixes carrier :: "'c set"   and more :: "'d"   and carrier' :: "'c set"   and more' :: "'d" shows "(\<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier', \<dots> = more'\<rparr>) = (carrier = carrier' \<and> more = more')"
rel_mono: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "R \<le> S" shows "rel R \<le> rel S"
assms: shows "\<Gamma> \<turnstile> M : \<tau>"
propex\<^sub>3: fixes F :: "\<Pi>\<^sub>3" shows "\<exists>r. Some r = Semantics.d\<^sub>3 F"
z_idem: shows "\<^bold>1 \<^bold>* \<^bold>1 = \<^bold>1"
assms_4: shows "dom G \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"
assms_2: shows "AE x in M. X x \<in> I"
disj_convr: fixes p :: "('c, 'b) urel"   and q :: "('c, 'b) urel" shows "(p \<or> q)\<^sup>- = (q\<^sup>- \<or> p\<^sup>-)"
eq_fst_iff: fixes a :: "'a"   and p :: "'a \<times> 'b" shows "(a = fst p) = (\<exists>b. p = (a, b))"
R_trans: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "R a b"   and "R b c" shows "R a c"
sg_condition: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<forall>a b. a \<in> H \<and> b \<in> H \<longrightarrow> a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b \<in> H" shows "G \<guillemotright> H "
R_lin_independent_Cons_imp_independent_RSpans: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b"   and ms :: "'b list" assumes "RModule R smult M"   and "m \<in> M"   and "R_scalar_mult.R_lin_independent R smult (m # ms)" shows "add_independentS [R_scalar_mult.RSpan R smult [m], R_scalar_mult.RSpan R smult ms]"
Rep_process: fixes x :: "'c process" shows "Rep_process x \<in> process_set"