hcju/isabelleps huggingface dataset:
add_mset_eq_single: fixes b :: "'a"   and M :: "'a multiset"   and a :: "'a" shows "(add_mset b M = \<LM> a  \<RM>) = (b = a \<and> M = \<Empt>)"
extensional_funcset_fun_upd_extends_rangeI: fixes a :: "'c"   and T :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and x :: "'d" assumes "a \<in> T"   and "f \<in> S \<rightarrow>\<^sub>E T - {a}" shows "f(x := a) \<in> insert x S \<rightarrow>\<^sub>E T"
sum_list_map_aezfun: fixes f :: "'c \<Rightarrow> ('b, 'a) aezfun"   and as :: "'c list" shows "aezfun (sum_list (map f as)) = (\<Sum>a\<leftarrow>as. aezfun (f a))"
correct_dims: fixes B :: "'a tensor" assumes "B \<in> set (map (\<lambda>i. unit_vec (hd (dims A)) i \<otimes> subtensor A i) [0..<hd (dims A)])" shows "dims B = dims A"
bottom_def: fixes L :: "('a, 'b) gorder_scheme" shows "\<bottom>\<^bsub>L\<^esub> = (SOME x. least L x (carrier L))"
saturate_commute: fixes r :: "'a"   and g :: "'a set \<Rightarrow> 'a set"   and h :: "'a set \<Rightarrow> 'a set"   and C :: "'a set" assumes "\<And>s. r \<in> g s"   and "\<And>s. g (insert r s) = g s"   and "\<And>s. r \<in> s \<Longrightarrow> h s = g s"   and "mono g"   and "\<And>X. X \<subseteq> C \<Longrightarrow> g X \<subseteq> C"   and "finite C" shows "saturate g {} = saturate h {r}"
assms_2: fixes a :: "name"   and y :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a<\<nu>y> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q"   and "y \<sharp> C" shows "\<exists>P'. P \<longmapsto> a<\<nu>y> \<prec> P' \<and> (P', Q') \<in> Rel"
simps_2: fixes x :: "int" shows "take_decr [x] = [x]"
add_mset_remove_trivial_eq: fixes N :: "'a multiset"   and a :: "'a" shows "(N = add_mset a (N - {#a#})) = (a \<in># N)"
Rep_mod: fixes x :: "'a" shows "Rep x mod int CARD('a) = Rep x"
slide_def: fixes as :: "'b list"   and t :: "'b tree"   and w :: "nat \<times> nat" shows "slide as t w = (let ts = atomics as (max (fst w) (Suc (r t))) (snd w); ts' = reusables t w in fold combine (ts @ ts') Leaf)"
border_def: fixes x :: "'b list"   and w :: "'b list" shows "x \<le>b w = (x \<le>p w \<and> x \<le>s w \<and> x \<noteq> w \<and> x \<noteq> \<epsilon>)"
toplevel_summands_derivs_Star: fixes xs :: "'b list"   and r :: "'b rexp" shows "toplevel_summands (derivs xs (Star r)) \<subseteq> {Star r} \<union> {Times (derivs ys r) (Star r) |ys. \<exists>zs. ys \<noteq> [] \<and> zs @ ys = xs}"
Bound: fixes a :: "subject"   and y :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q'"   and "y \<sharp> P" shows "\<exists>P'. P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' Q' a y Rel"
hyps_3: shows "card {x \<in> Pow F. card x = 1} = card F"
blackD_1: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "black r s" shows "marked r s"
sort_all_coeffs_gen: fixes xs' :: "(char list list \<times> int) list"   and xs :: "(char list list \<times> int) list" assumes "\<forall>xs\<in>mononoms xs'. sorted_wrt var_order xs"   and "\<forall>x\<in>mononoms (xs @ xs'). distinct x" shows "monadic_nfoldli xs (\<lambda>_. RETURN True) (\<lambda>(a, n) b. sort_coeff a \<bind> (\<lambda>a. RETURN ((a, n) # b))) xs' \<le> \<Down> Id (SPEC (\<lambda>ys. map (\<lambda>(a, b). (mset a, b)) (rev xs @ xs') = map (\<lambda>(a, b). (mset a, b)) ys \<and> (\<forall>xs\<in>mononoms ys. sorted_wrt var_order xs)))"
inject_7: fixes x1 :: "('d, 'e, 'f) psi"   and y1 :: "('d, 'e, 'f) psi" shows "(!x1 = !y1) = (x1 = y1)"
two_orders: fixes x :: "'a"   and y :: "'a" assumes "domain_semigroup.ds_ord ans_d (\<cdot>) x y" shows "x \<le> y"
2: shows "rword_strict (spp_of b) (spp_of b')"
multiset_image_do_nothing: fixes A :: "'b multiset"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in># A \<Longrightarrow> f x = x" shows "image_mset f A = A"
sunflower_nonempty_core_lift: fixes E :: "'b set"   and k :: "nat"   and c :: "nat"   and r :: "nat"   and F :: "'b set set" assumes "finite E"   and "\<And>G. \<lbrakk>\<forall>A\<in>G. finite A \<and> card A \<le> k; c < card G\<rbrakk> \<Longrightarrow> \<exists>S\<subseteq>G. sunflower S \<and> card S = r"   and "\<forall>A\<in>F. A \<subseteq> E \<and> card A \<le> k"   and "{} \<notin> F"   and "card E * c < card F" shows "\<exists>S\<subseteq>F. sunflower S \<and> card S = r \<and> \<Inter> S \<noteq> {}"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = f\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f"
quicksort_term1: fixes r :: "nat"   and l :: "nat"   and xs :: "'b list"   and x :: "nat \<times> 'b list"   and p :: "nat"   and xs1 :: "'b list" assumes "\<not> r \<le> l"   and "\<not> length xs \<le> r"   and "x = Quicksort.partition xs l r"   and "(p, xs1) = x" shows "p - Suc l < r - l"
sub_designI: fixes \<U> :: "'a set"   and \<A> :: "'a set multiset" assumes "design \<U> \<A>"   and "sub_incidence_system \<U> \<A> \<V> \<B>" shows "sub_design \<U> \<A> \<V> \<B>"
bounded_linear_minus: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "bounded_linear (\<lambda>x. - f x)"
word_log2_nth_not_set: fixes w :: "'b word"   and i :: "nat" assumes "word_log2 w < i"   and "i < size w" shows "\<not> w !! i"
ereal_le_le: fixes y :: "real"   and a :: "ereal"   and x :: "real" assumes "ereal y \<le> a"   and "x \<le> y" shows "ereal x \<le> a"
ccw'_sortedP_polychain_of_snd: fixes xs :: "(real \<times> real) list"   and x0 :: "real \<times> real" assumes "ccw'.sortedP 0 xs" shows "ccw'.sortedP x0 (map snd (polychain_of x0 xs))"
calculus_axioms: shows "calculus Bot_FL Inf_FL (\<lambda>N1 N2. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2))) ord_q_lifting.Red_I_\<G> local.ord_q_lifting.Red_F_\<G>"
dual_equivalence: shows "internal_equivalence d\<^sub>1 e\<^sub>1 (local.inv \<epsilon>\<^sub>1) (local.inv \<eta>\<^sub>1)"
Throw: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "nrm A = UNIV"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>Throw e\<rangle>\<guillemotright> A"
set_iterator_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set" shows "set_iterator it S0 \<equiv> set_iterator_genord it S0 (\<lambda>_ _. True)"
zero_pratt2_def: shows "0 \<equiv> \<bottom>\<^sub>2"
transfer_operations_rl_1: fixes xrs :: "real list"   and yrs :: "real list"   and xri :: "real list"   and yri :: "real list" assumes "SIDE_PRECOND (list_all2 (\<le>) xrs yrs)"   and "(xri, xrs) \<in> rl_rel"   and "(yri, yrs) \<in> rl_rel" shows "(appr_of_ivl ops xri yri, lv_ivl $ xrs $ yrs) \<in> appr_rell"
to_vec_inject: fixes x :: "'a sq_mtx"   and y :: "'a sq_mtx" shows "(to_vec x = to_vec y) = (x = y)"
get_mode_ok: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr" assumes "ShadowRootClass.type_wf h"   and "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_mode shadow_root_ptr"
that: fixes v :: "'a llist"   and a :: "'a" assumes "w = v $ <a>"   and "llength v = enat n" shows "thesis"
elims_3: fixes x :: "value" assumes "\<not> is_Num x" obtains uv_ :: "String.literal" where "x = Str uv_"
sums_integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'a" assumes "integrable (count_space UNIV) f" shows "f sums integral\<^sup>L (count_space UNIV) f"
circ_right_top: fixes x :: "'a" shows "times_top_class.total (x\<^sup>\<star>)"
insert_remove: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "mset_set (insert x A) = add_mset x (mset_set (A - {x}))"
length_ge_one: fixes x :: "'a list" shows "x \<noteq> [] \<longrightarrow> 1 \<le> length x"
Pi_mem: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'h \<Rightarrow> 'i set"   and x :: "'h" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
vconst_onE: fixes x :: "V"   and A :: "V"   and c :: "V" assumes "x \<in>\<^sub>\<circ> vconst_on A c" obtains a :: "V" where "a \<in>\<^sub>\<circ> A" and "x = \<langle>a, c\<rangle>"
exec_move_def: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1" shows "exec_move ci P t e \<equiv> exec_meth ci (compP2 P) (compE2 e) (compxE2 e 0 0) t"
alg_closed_imp_poly_has_root: fixes p :: "'b poly" assumes "0 < degree p" shows "\<exists>x. poly p x = (0::'b)"
1: shows "transp (\<lambda>a\<^sub>1 a\<^sub>2. P a\<^sub>1 \<longrightarrow> P a\<^sub>2)"
simps_3: fixes par :: "ty"   and res :: "ty"   and lcl :: "(vname \<times> ty) list"   and bdy :: "stmt"   and more :: "'a" shows "lcl \<lparr>par = par, res = res, lcl = lcl, bdy = bdy, \<dots> = more\<rparr> = lcl"
assms: shows "u \<rightarrow>\<^bsub>G \<restriction> S\<^esub> v"
sst_vars_proj_subset_3: fixes n :: "'g"   and A :: "('g strand_label \<times> ('f, 'e) stateful_strand_step) list" shows "vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> vars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
regions_closed': fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and R :: "('d \<Rightarrow> real) set"   and v :: "'d \<Rightarrow> real"   and t :: "real" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "R \<in> \<R>"   and "v \<in> R"   and "0 \<le> t" shows "v \<oplus> t \<in> [v \<oplus> t]\<^sub>\<R>"
unfold: shows "subst_typ (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_typ (zip (map fst insts) (map2 Tv fresh_idns (map snd (map fst insts)))) \<tau>) = fold (\<lambda>single. subst_typ [single]) (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (fold (\<lambda>single. subst_typ [single]) (zip (map fst insts) (map2 Tv fresh_idns (map snd (map fst insts)))) \<tau>)"
obs_neq_some_contradicts: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "obs \<sigma> k \<noteq> obs \<sigma>' k"   and "obs \<sigma> k \<noteq> None"   and "obs \<sigma>' k \<noteq> None" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> path \<sigma> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> path \<sigma>' k' \<in> dom local.att"
not_outside_2: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "Order_Predicates.preorder_on carrier le"   and "le x y" shows "y \<in> carrier"
mon_env_ileq: fixes w :: "('a, 'b) label list el_step list"   and w' :: "('a, 'b) label list el_step list"   and fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" assumes "w \<preceq> w'" shows "mon_env fg w \<subseteq> mon_env fg w'"
that: fixes Q_hit :: "'a list"   and Q_hit_pre :: "'a list"   and Q_hit_post :: "'a list" assumes "Q_hit \<in> Q"   and "y \<in> set Q_hit"   and "Q_hit = Q_hit_pre @ y # Q_hit_post" shows "thesis"
length_ode_fa: fixes xs :: "floatarith list" shows "length (ode_fa xs) = length ode_e"
cs_sym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "relation.cs r x y" shows "relation.cs r y x"
\<delta>_inj_on_t: shows "inj_on \<delta> (fv t)"
inv_into_into: fixes x :: "'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
rev_iterateoi_correct: fixes S :: "'s" assumes "invar S" shows "map_iterator_rev_linord (reverse_iterateoi S) (\<alpha> S)"
funpow_dist_0_eq: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "(funpow_dist f x y = 0) = (x = y)"
vars_\<C>_mono: fixes X :: "'Var set"   and Y :: "'Var set" assumes "X \<subseteq> Y" shows "vars_\<C> X \<subseteq> vars_\<C> Y"
terminates_def: fixes g :: "'e \<Rightarrow> ('f, 'e) step" shows "terminates g = (terminates_on g = UNIV)"
simps_1: fixes x :: "'a" shows "norm (Opaque x) = Pure \<I> \<diamondop> Opaque x"
Equal: shows "{sat_models (Abs \<phi>) |\<phi>. nested_propos \<phi> \<subseteq> P} = {{A \<union> B |A B. A \<subseteq> P \<and> A \<Turnstile>\<^sub>P \<phi> \<and> B \<subseteq> UNIV - P} |\<phi>. nested_propos \<phi> \<subseteq> P}"
abs_eq: fixes xa :: "perm"   and x :: "('e, 'f, 'g) Tree" shows "xa \<bullet> abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
iter_tensor_mult_distr: fixes n :: "nat"   and A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" assumes "1 \<le> n"   and "dim_col A = dim_row B"   and "0 < dim_col A"   and "0 < dim_col B" shows "A \<otimes>\<^bsup>Suc n\<^esup> * B \<otimes>\<^bsup>Suc n\<^esup> = A * B \<Otimes> A \<otimes>\<^bsup>n\<^esup> * B \<otimes>\<^bsup>n\<^esup>"
pol_coeff_mem_R: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and j :: "nat" assumes "pol_coeff S c"   and "j \<le> fst c" shows "snd c j \<in> carrier R"
induct: fixes x1 :: "'e list"   and x2 :: "'f list"   and R :: "('e \<times> 'f) set"   and P :: "'e list \<Rightarrow> 'f list \<Rightarrow> bool" assumes "(x1, x2) \<in> seq_lift R"   and "P [] []"   and "\<And>xs ys x y. \<lbrakk>(xs, ys) \<in> seq_lift R; P xs ys; (x, y) \<in> R\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
eq1_1: shows "lst \<sigma>1' = fst \<sigma>2'"
induced_subgraph_altdef2: fixes H :: "('a, 'b) pre_digraph" shows "induced_subgraph H G = max_subgraph (\<lambda>H'. verts H' = verts H) H"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C v1, $C v2, $C vn, $e] : ts _> ts'"
update_conv': fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "map_of (update k v al) = map_of al(k \<mapsto> v)"
upper_density_delta: fixes d :: "real" assumes "0 < d"   and "d \<le> 1" shows "\<exists>delta. (\<forall>l. 0 < delta l) \<and> delta \<longlonglongrightarrow> 0 \<and> ennreal (1 - d) < emeasure M {x \<in> space M. \<forall>N. real (card {n \<in> {..<N}. \<exists>l\<in>{1..n}. u n ((Tinv ^^ n) x) - u (n - l) ((Tinv ^^ (n - l)) x) \<le> - delta l * real l}) \<le> d * real N}"
mkProc_pState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and pidN :: "nat"   and sidx :: "nat"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list" assumes "program_inv prog"   and "gState_inv prog g"   and "pidN \<le> max_procs"   and "0 < pidN"   and "sidx < IArray.length (processes prog)"   and "fst (processes prog !! sidx) = sidx" shows "pState_inv prog (snd (mkProc g p name args (processes prog !! sidx) pidN))"
neg_bounds_compare_4: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c <\<^sub>u\<^sub>b b" shows "c \<ge>\<^sub>u\<^sub>b b"
cases: fixes a1 :: "'b"   and a2 :: "'c"   and r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" assumes "(a1, a2) \<in> r O s" obtains a :: "'b" and b :: "'d" and c :: "'c" where "a1 = a" and "a2 = c" and "(a, b) \<in> r" and "(b, c) \<in> s"
foldr_conv_fold: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f list" shows "foldr f xs = fold f (rev xs)"
map\<^sub>0_simps_4: fixes a :: "'a" assumes "C.obj a" shows "D.dom (G.map\<^sub>0 a) = G.map\<^sub>0 a"
lex_comp'_pp: fixes x :: "('a, 'b) pp"   and y :: "('a, 'b) pp" shows "lex_comp' x y = comp_of_ord lex_pp (rep_nat_pp x) (rep_nat_pp y)"
zeros_def: fixes m :: "nat"   and n :: "nat" shows "zeros m n = replicate m (replicate n False)"
tr_par_preserves_par_comp: fixes A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list" assumes "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"   and "A' \<in> set (tr\<^sub>p\<^sub>c A [])" shows "par_comp A' Sec"
bit_set_bit_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and m :: "nat"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_bit_operations.set_bit one push_bit or m a) n = (bit a n \<or> m = n \<and> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
strictly_subsumes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.strictly_subsumes subst_atm C D = strict (substitution_ops.subsumes subst_atm) C D"
reduced_children_parent: fixes l :: "dag"   and llt :: "dag"   and lp :: "ref"   and rlt :: "dag"   and r :: "dag"   and lrt :: "dag"   and rp :: "ref"   and rrt :: "dag"   and p :: "ref" assumes "reduced l"   and "l = Node llt lp rlt"   and "reduced r"   and "r = Node lrt rp rrt"   and "lp \<noteq> rp" shows "reduced (Node l p r)"
plus_absorb1: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "O[F](\<lambda>x. f x + g x) = O[F](g)"
subformulas\<^sub>\<mu>_list_set: fixes \<phi> :: "'b ltln" shows "set (subformulas\<^sub>\<mu>_list \<phi>) = subformulas\<^sub>\<mu> \<phi>"
funpow_Suc_conv: fixes n :: "nat"   and m :: "nat" shows "(Suc ^^ n) m = m + n"
o_bij: fixes g :: "'i \<Rightarrow> 'h"   and f :: "'h \<Rightarrow> 'i" assumes "g \<circ> f = id"   and "f \<circ> g = id" shows "bij f"
wadjust_on_left_moving_B_Bk2: fixes m :: "nat"   and rs :: "nat"   and n :: "nat" shows "wadjust_on_left_moving_B m rs (Bk \<up> n @ Bk # Oc # Oc # Oc \<up> rs @ Bk # Oc # Oc \<up> m, [Bk])"
add_rat_root_cond: fixes m :: "rat"   and p :: "int poly"   and l :: "rat"   and r :: "rat"   and x :: "real" shows "root_cond (cf_pos_poly (poly_add_rat m p), l, r) x = root_cond (p, l - m, r - m) (x - real_of_rat m)"
pull_up_senior_older_seniors: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n" shows "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n"
G_def: shows "residue_mult_group d \<equiv> residue_mult_group d"
disc_2: fixes A :: "'c llist set" assumes "\<not> (\<forall>x\<in>A. lnull x)" shows "\<not> lnull (lSup A)"
distinct_45: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x91 :: "'d"   and x92 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x93 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x91}receive(x92) . x93"
matrix_mul_assoc: fixes A :: "(('d, 'g) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and B :: "(('d, 'h) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec"   and C :: "(('d, 'e) Finite_Cartesian_Product.vec, 'h) Finite_Cartesian_Product.vec" shows "A ** (B ** C) = A ** B ** C"
subset_mono: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b set" assumes "A \<subseteq> B" shows "M \<subseteq> A \<longrightarrow> M \<subseteq> B"
leg1_simps_2: shows "hseq (tab\<^sub>1 f) f\<^sub>0gh\<^sub>1.p\<^sub>1"
apex_composite: shows "trg_f.apex = trg.leg0 \<down>\<down> f.leg1"
finite_lin_span: fixes A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and n1 :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "h \<in> {j. j \<le> n} \<rightarrow> carrier M"   and "s \<in> {j. j \<le> n1} \<rightarrow> A"   and "f \<in> {j. j \<le> n1} \<rightarrow> h ` {j. j \<le> n}" shows "\<exists>t\<in>{j. j \<le> n} \<rightarrow> A. l_comb R M n1 s f = l_comb R M n t h"
span_scale: fixes x :: "'d"   and S :: "'d set"   and c :: "complex" assumes "x \<in> cspan S" shows "c *\<^sub>C x \<in> cspan S"
mult_minus_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "times a (uminus b) = uminus (times a b)"
degree_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "(f (lead_coeff p) = (0::'a)) = (p = 0)" shows "degree (map_poly f p) = degree p"
rotate_inv1: fixes n :: "nat"   and ls :: "'c list"   and ms :: "'c list" assumes "n < |ls|"   and "rotate n ls = ms" shows "ls = rotate (|ls| - n) ms"
is_Atom_def: fixes rexp :: "'b rexp" shows "is_Atom rexp = (\<exists>x3. rexp = Atom x3)"
x: shows "x \<in> {(hd vs, ram1), (ram1, hd vs1), (last vs1, ram2), (ram2, last vs)} \<union> Edges (rev vs) \<union> Edges vs1"
simp': shows "subst_bv (Fv x \<tau>) (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0)) = mk_eq' \<tau>' (f $ Fv x \<tau>) (g $ Fv x \<tau>)"
valid_eval_match: fixes env :: "v sem_env"   and s :: "'a state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" shows "evaluate_match True env s v pes err_v (eval_match env v pes err_v s)"
degree_normalize: fixes f :: "'a poly" shows "degree (normalize f) = degree f"
id2: fixes f :: "nat \<Rightarrow> 'b" shows "f k # rev (map f [0..<k]) = rev (map f [0..<Suc k])"
distinct_1: fixes x1 :: "('e \<times> 'f) pmf"   and x2 :: "('e \<times> 'f) set['g]" shows "PMF x1 \<noteq> BPS x2"
conjugate_vec_def: fixes v :: "'a vec" shows "conjugate v = Matrix.vec (dim_vec v) (\<lambda>i. conjugate (v $ i))"
dim_vec_last: fixes v :: "'b Matrix.vec"   and n :: "nat" shows "dim_vec (vec_last v n) = n"
fixed_point_imp_closed_orbit_period_zero_2: fixes x :: "'a" assumes "x \<in> X"   and "f x = (0::'a)" shows "period x = 0"
functor_axioms: shows "functor J' C (D \<circ> F)"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case EV_PREP of EV_PREP \<Rightarrow> f1 | EV_WAIT \<Rightarrow> f2 | EV_FINISH \<Rightarrow> f3) = f1"
stl_sset: fixes x :: "'a"   and a :: "'a stream" assumes "x \<in> i.R (stl a)" shows "x \<in> i.R a"
DEFL_app: shows "DEFL('c\<cdot>'d) = TC('d)\<cdot>DEFL('c)"
h: fixes T :: "real"   and x :: "nat \<Rightarrow> real" assumes "0 \<le> T"   and "T \<le> 1"   and "(\<Sum>i\<le>n. (x i)\<^sup>2) = 1"   and "\<forall>i>n. x i = 0"   and "- T \<le> x k" shows "h (T, x) = x"
path_edge: fixes l1' :: "'loc"   and lbl :: "'sum"   and l2' :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and l1 :: "'loc"   and l2 :: "'loc" assumes "(l1', lbl, l2') \<in> set xs"   and "path l1 l2 xs" shows "lbl \<in>\<^sub>A summary l1' l2'"
distinct_19: fixes x5 :: "loc" shows "Null \<noteq> Addr x5"
splits_at_combine: fixes \<delta> :: "symbol list"   and i :: "nat"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "splits_at \<delta> i \<alpha> N \<beta>" shows "\<delta> = \<alpha> @ [N] @ \<beta>"
fv_Cons: fixes x :: "'b"   and xs :: "'b list" shows "fv (x # xs) = fv x \<union> fv xs"
subtuple_antisym: fixes ym :: "'b \<rightharpoonup>\<^sub>f 'c"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xm :: "'b \<rightharpoonup>\<^sub>f 'c" assumes "subtuple (\<lambda>x y. f x y \<or> x = y) xm ym"   and "subtuple (\<lambda>x y. f x y \<and> \<not> f y x \<or> x = y) ym xm" shows "xm = ym"