hcju/isabelleps huggingface dataset:
sinv: shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) Q"
transfer: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k" assumes "class.ring_1 one times plus zero minus uminus" shows "rel_fun pcr_word (=) (ring_1.of_int one plus zero minus \<circ> signed_take_bit (LENGTH('l) - Suc 0)) (ring_1.signed one plus zero minus)"
2: shows "H * a * top \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
cond_disj_distr: fixes P :: "'a upred"   and b :: "'a upred"   and Q :: "'a upred"   and S :: "'a upred" shows "(P \<or> Q \<triangleleft> b \<triangleright> S) = (P \<or> Q) \<triangleleft> b \<triangleright> (P \<or> S)"
Abs_group_inject: fixes x :: "'d monoid"   and y :: "'d monoid" assumes "x \<in> {G. Group.group G}"   and "y \<in> {G. Group.group G}" shows "(Abs_group x = Abs_group y) = (x = y)"
par_skip: fixes par :: "'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice"   and skip :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice" assumes "parallel (par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (skip::'a::refinement_lattice)" shows "(par::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (a::'a::refinement_lattice) (skip::'a::refinement_lattice) = a"
1: shows "(f has_integral set_lebesgue_integral lborel S f) S"
weak_complete_latticeI: assumes "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>s. is_lub L s A"   and "\<And>A. A \<subseteq> carrier L \<Longrightarrow> \<exists>i. is_glb L i A" shows "weak_complete_lattice L"
append_eq_append_conv2: fixes xs :: "'f list"   and ys :: "'f list"   and zs :: "'f list"   and ts :: "'f list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
IH: shows "(p + q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r + q \<circ>\<^sub>p r"
assoc_list_ext: fixes xs :: "('b, 'c) assoc_list"   and ys :: "('b, 'c) assoc_list" assumes "assoc_list.impl_of xs = assoc_list.impl_of ys" shows "xs = ys"
bin_last_set_bits: fixes f :: "nat \<Rightarrow> bool" assumes "wf_set_bits_int f" shows "odd (set_bits f) = f 0"
eq_Abs_Exp: fixes z :: "exp" obtains U :: "freeExp" where "z = Abs_Exp (exprel `` {U})"
obj_simps_1: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.arr a"
ct_prefixI: fixes ys :: "ct list"   and as :: "ct list"   and zs :: "ct list"   and xs :: "ct list" assumes "ys = as @ zs"   and "ct_list_eq as xs" shows "ct_prefix xs ys"
wf_weaken': fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T"
q2: shows "q2 = hom_component p (deg_pm s2)"
extensions_def: shows "Id.R.extensions \<equiv> {La. field La \<and> Id.R.indexed_const \<in> ring_hom (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>) La \<and> Ball (carrier La) Id.R.carrier_coeff \<and> (\<forall>\<P>\<in>carrier La. \<forall>P\<in>carrier (poly_ring (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)). \<forall>i. \<not> Id.R.index_free \<P> (P, i) \<longrightarrow> Id.R.indexed_var (P, i) \<in> carrier La \<and> ring.eval La (Id.R.\<sigma> P) (Id.R.indexed_var (P, i)) = \<zero>\<^bsub>La\<^esub>)}"
cr_rename_gba_def: fixes nds :: "('a, 'b) node_scheme set"   and \<phi> :: "'a ltlr" shows "cr_rename_gba nds \<phi> \<equiv> let V = name ` nds; V0 = name ` {q \<in> nds. expand_init \<in> incoming q} in build_succ nds \<bind> (\<lambda>succmap. let E = E_of_succ (the_default {} \<circ> succmap); F = build_F nds \<phi> in pn_map nds \<bind> (\<lambda>pnm. let L = \<lambda>qn l. case pnm qn of None \<Rightarrow> False | Some (P, N) \<Rightarrow> (\<forall>p\<in>P. p \<in> (l ::: \<langle>Id\<rangle>fun_set_rel)) \<and> (\<forall>p\<in>N. p \<notin> l) in RETURN \<lparr>g_V = V, g_E = E, g_V0 = V0, gbg_F = F, gba_L = L\<rparr>))"
ev_blocks_part_leD: fixes m :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "ev_blocks_part m A"   and "i \<le> j"   and "j \<le> k"   and "k < m"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"
synth_analz_snd: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> synth (analz H)" shows "Y \<in> synth (analz H)"
set_cases: fixes e :: "'b"   and a :: "'b err" assumes "e \<in> set_err a" obtains "a = OK e"
distinct_5: fixes x2 :: "jump"   and x3 :: "error" shows "Jump x2 \<noteq> Error x3"
size_add_elem_step_eq: fixes A :: "'a multiset"   and B :: "'b multiset"   and x :: "'a" assumes "size (A \<times># B) = size A * size B" shows "size (add_mset x A \<times># B) = size (add_mset x A) * size B"
extensional_restrict: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f \<in> extensional A" shows "restrict f A = f"
multiplicative_sum_divisors: shows "multiplicative_function (\<lambda>n. \<Sum>d | d dvd n. liouville_lambda d)"
interaction_bounded_by_Done: fixes "consider" :: "'b \<Rightarrow> bool"   and x :: "'d"   and n :: "enat" shows "interaction_bounded_by consider (Generative_Probabilistic_Value.Done x) n"
assms: fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "\<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge V V' (x, \<rho>)) e)) \<partial>state_measure V \<Gamma> \<le> 1"
mult_sheaf_spec_in_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and U :: "'c set set"   and t :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit"   and "s \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"   and "t \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"   and "U \<subseteq> comm_ring.spectrum R addition multiplication zero unit" shows "comm_ring.mult_sheaf_spec R addition multiplication zero U s t \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
set_dom_cod_object_subset_1: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "Dom\<^bsub>set_cat U\<^esub> f \<in> Ob\<^bsub>set_cat U\<^esub>"
nth_Basis_list_prod: fixes i :: "nat" assumes "i < DIM('a) + DIM('b)" shows "Basis_list ! i = (if i < DIM('a) then (Basis_list ! i, 0::'b) else (0::'a, Basis_list ! (i - DIM('a))))"
acc_induct: fixes a :: "'g"   and r :: "('g \<times> 'g) set"   and P :: "'g \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<forall>y. (y, x) \<in> r \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
sum_SDG_call_summary_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a')" shows "n s-p\<rightarrow>\<^bsub>sum\<^esub> n'"
dominates_antisymm': fixes g :: "'g"   and n :: "'node"   and n' :: "'node" assumes "old.dominates g n n'"   and "old.dominates g n' n" shows "n = n'"
M_1: shows "\<forall>l. \<forall>m\<in>M l. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec"
inner_node_def: fixes n :: "'node" shows "inner_node n \<equiv> valid_node n \<and> n \<noteq> (_Entry_) \<and> n \<noteq> (_Exit_)"
Pareto_iff: fixes x :: "'b"   and y :: "'b" shows "Pareto R x y = (\<forall>i\<in>agents. R i x y)"
semi_forgetful_RO_def: shows "semi_forgetful_RO = (\<lambda>seed (\<sigma>, b) x. case \<sigma> (h seed x) of None \<Rightarrow> rand \<bind> (\<lambda>y. return_spmf (y, \<sigma>(h seed x \<mapsto> (x, y)), b)) | Some (a, y) \<Rightarrow> return_spmf (y, \<sigma>, a \<noteq> x \<or> b))"
intro: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "principalideal_axioms I R" shows "principalideal I R"
weakPsiCongParPresSym: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi>. \<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> R \<parallel> P \<doteq> R \<parallel> Q"
Im_linepath': fixes a :: "complex"   and b :: "complex"   and x :: "real" shows "Im (linepath a b x) = linepath (Im a) (Im b) x"
nn_integral_measure_pmf_finite: fixes M :: "'b pmf"   and f :: "'b \<Rightarrow> ennreal" assumes "finite (set_pmf M)"   and "\<And>x. x \<in> set_pmf M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>N (measure_pmf M) f = (\<Sum>x\<in>set_pmf M. f x * ennreal (pmf M x))"
is_rcoset_replistD_cosets: fixes H :: "'g set"   and gs :: "'g list" assumes "is_rcoset_replist H gs" shows "G = (\<Union>g\<in>set gs. rcoset_rel H `` {g})"
help''': fixes ram :: "'a"   and xs :: "'a list"   and b :: "'a list" shows "fst (splitAtRec ram xs b) = xs @ fst (splitAtRec ram [] b)"
ty_fresh: fixes x :: "name"   and T :: "ty" shows "x \<sharp> T"
Ord_induct: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord k"   and "\<And>x. \<lbrakk>Ord x; \<And>y. y \<in>\<^sub>\<circ> x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P k"
wf_ll_def: fixes t :: "dag"   and levellist :: "ref list list"   and var :: "ref \<Rightarrow> nat" shows "wf_ll t levellist var = ((\<forall>p. p \<in> set_of t \<longrightarrow> p \<in> set (levellist ! var p)) \<and> (\<forall>k<length levellist. \<forall>p\<in>set (levellist ! k). p \<in> set_of t \<and> var p = k))"
l3_inv10_init: shows "init l3 \<subseteq> l3_inv10"
simplex_extremal_le: fixes S :: "'b set" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>u\<in>S. \<exists>v\<in>S. \<forall>x\<in>convex hull S. \<forall>y\<in>convex hull S. norm (x - y) \<le> norm (u - v)"
cons_set_eq: fixes x :: "'b"   and xs :: "'b list" shows "set (x # xs) \<inter> set xs = set xs"
obtuse__nsams: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Obtuse A B C" shows "\<not> SAMS A B C A B C"
NMPcharn: fixes a :: "'b"   and p :: "'b list"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "a \<in> set p"   and "dom (C a) \<noteq> {}" shows "not_MT C p"
Call: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and bdy :: "('a, 'b, 'c) com"   and s :: "'a"   and t :: "('a, 'c) xstate" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> t"
PX3d_linear: fixes d :: "D2" shows "Linear_order (PX3d d)"
0: shows "a \<in> injections Y N \<and> Y \<in> all_partitions G"
hyps: fixes w :: "'a stream"   and thesis :: "bool" assumes "shd w < shd wa__"   and "\<And>u a. w = u @- sconst a \<Longrightarrow> thesis"   and "sdescending w" shows "thesis"
Rep_scene_inject: fixes x :: "'a scene"   and y :: "'a scene" shows "(Rep_scene x = Rep_scene y) = (x = y)"
flip_altsublist_adjacent_set: fixes ss :: "'w list"   and ts :: "'w list" assumes "ss \<in> lists S"   and "flip_altsublist_adjacent ss ts" shows "set ts = set ss"
elims_2: fixes x :: "('d \<times> 'e \<times> 'd) set \<times> 'd \<times> (('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set"   and xa :: "nat \<Rightarrow> 'e" assumes "accept\<^sub>G\<^sub>R_LTS x xa" obtains \<delta> :: "('d \<times> 'e \<times> 'd) set" and q\<^sub>0 :: "'d" and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set" and w :: "nat \<Rightarrow> 'e" where "x = (\<delta>, q\<^sub>0, \<alpha>)" and "xa = w" and "\<exists>(Fin, Inf)\<in>\<alpha>. accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 (Fin, Inf) w"
eval\<^sub>G_Or_map: fixes S :: "'a ltl set"   and xs :: "'a ltl list" shows "eval\<^sub>G S (Or xs) = Or (map (eval\<^sub>G S) xs)"
countable2: fixes n :: "nat" shows "countable (\<^bold>E `` A_n n)"
finite_vars_imp_finite_adapt_b: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" assumes "finite (Bexp.vars e)" shows "finite (Bexp.vars (adapt_bexp e s))"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> orb" shows "X \<in> knows Spy evs"
has_integral_cmul: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and S :: "'a set"   and c :: "real" assumes "(f has_integral k) S" shows "((\<lambda>x. c *\<^sub>R f x) has_integral c *\<^sub>R k) S"
permute_var_def: fixes p :: "perm"   and a :: "Atoms.var" shows "p \<bullet> a = Abs_var (p \<bullet> Rep_var a)"
strict_assoc: fixes a0 :: "'d"   and a1 :: "'d"   and a2 :: "'d" assumes "D.ide a0"   and "D.ide a1"   and "D.ide a2" shows "D.ide \<a>\<^sub>D[a0, a1, a2]"
Quotient_Arity: shows "Quotient (=) to_Arity Rep_Arity cr_Arity"
weak_reduction_correspondence_simulation_impl_coupled_simulation_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_correspondence_simulation Rel Cal" shows "\<forall>P Q. (P, Q) \<in> Rel \<longrightarrow> (P, Q) \<in> cSim_cs Rel Cal \<and> (Q, P) \<in> cSim_cs Rel Cal"
that: fixes U' :: "'a set"   and V :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "open U'"   and "U' \<subseteq> U"   and "(0::'a) \<in> U'"   and "open V"   and "(0::'a) \<in> V"   and "homeomorphism U' V f g"   and "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative g' y) (at y)"   and "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f' (g y)))"   and "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f' (g y)))" shows "thesis"
assms: shows "makespan (T(x := T x + y)) \<noteq> T x + y"
trans_P2R: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "trans (P2R \<phi>) = transP \<phi>"
I_eval: fixes x :: "dB" shows "\<I> \<degree> x \<rightarrow>\<^sub>\<beta> x"
valid_plan_mems: fixes as :: "'d list"   and PROB :: "'d set"   and a :: "'d" assumes "as \<in> valid_plans PROB"   and "ListMem a as" shows "a \<in> PROB"
adjoint_pair_antipar_2: fixes f :: "'a"   and g :: "'a" assumes "adjoint_pair f g" shows "ide g"
GsT: shows "(mat_of_rows n (map gs1.gso [0..<m]))\<^sup>T \<in> carrier_mat n m"
flv_ref_rel_def: shows "flv_ref_rel = {(sa, sc). sc = \<lparr>opt_v_state.next_round = v_state.next_round sa, last_vote = map_option snd \<circ> process_mru (votes sa), decisions = v_state.decisions sa\<rparr>}"
Un_Diff_Int: fixes A :: "'d set"   and B :: "'d set" shows "A - B \<union> A \<inter> B = A"
pred_fmap_fmpred: fixes P :: "'d \<Rightarrow> bool" shows "pred_fmap P = fmpred (\<lambda>_. P)"
axioms_5: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "natural_transformation A B F G \<tau>" shows "natural_transformation_axioms A B F G \<tau>"
assms_1: fixes S :: "'a set" assumes "S \<in> \<F>" shows "compact S"
paval_aval2: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "vars a \<subseteq> dom ps"   and "ps \<preceq> part s" shows "paval a ps = aval a s"
bij_lens_vwb: fixes x :: "'d \<Longrightarrow> 'e" assumes "bij_lens x" shows "vwb_lens x"
simps_2: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and a :: "'d" shows "map_sum f1 f2 (Inr a) = Inr (f2 a)"
flip_altsublist_adjacent_Cons_grow: fixes ss :: "'w list"   and ts :: "'w list"   and a :: "'w" assumes "flip_altsublist_adjacent ss ts" shows "flip_altsublist_adjacent (a # ss) (a # ts)"
WHILEIT_le_WHILEI: fixes V :: "('b \<times> 'b) set"   and I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and s :: "'b" assumes "wf V"   and "\<And>s. \<lbrakk>I s; b s; f s \<le> SPEC I\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. (s', s) \<in> V)" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s \<le> WHILE\<^bsup>I\<^esup> b f s"
csk: shows "cs\<^bsup>\<pi> \<guillemotleft> j\<^esup> k = cs\<^bsup>\<pi>' \<guillemotleft> j'\<^esup> k'"
LLL_inv_N_pos: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs"   and "g_bound fs"   and "m \<noteq> 0" shows "0 < N"
LeftDerivationFix_splits_at_nonterminal: fixes a :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and b :: "symbol list" assumes "LeftDerivationFix a i D j b"   and "is_nonterminal (a ! i)" shows "\<exists>U a1 a2 b1. splits_at a i a1 U a2 \<and> splits_at b j b1 U a2 \<and> LeftDerivation a1 D b1"
length_into_WW: fixes x :: "nat"   and ns :: "nat list" shows "length (into_WW x ns) = length ns"
preprocess'_unsat_index_farkas_coefficients_ns: fixes i :: "'a"   and ns :: "('a \<times> QDelta ns_constraint) list"   and j :: "nat" assumes "i \<in> set (UnsatIndices (preprocess' ns j))" shows "\<exists>C. farkas_coefficients_ns (snd ` set ns) C"
x_power_minus_1: fixes x :: "'b"   and n :: "nat" shows "x + (2::'b) ^ n - (1::'b) = x + ((2::'b) ^ n - (1::'b))"
le_iff_sup: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "less_eq_prod x y = (sup_prod x y = y)"
Q5_a_def: shows "Q5_a \<equiv> [: X, mrk \<leadsto> X', mrk' . X = {} \<and> mrk = mrk' :]"
f: shows "(f \<longlongrightarrow> g a) (at_right a)"
hom_add: fixes x :: "int"   and y :: "int" shows "of_int (x + y) = of_int x + of_int y"
list_update_beyond: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b" assumes "length xs \<le> i" shows "xs[i := x] = xs"
lless_def: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "(x \<sqsubset>\<^bsub>L\<^esub> y) = (x \<sqsubseteq>\<^bsub>L\<^esub> y \<and> x .\<noteq>\<^bsub>L\<^esub> y)"
induced_function_integral_nonergodic_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "conservative_mpt M T"   and "A \<in> sets M"   and "integrable M f" shows "integrable M (qmpt.induced_function T A f)"
insert_subset: fixes x :: "'k"   and A :: "'k set"   and B :: "'k set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
msgrel_imp_eqv_freeleft: fixes U :: "freemsg"   and V :: "freemsg" assumes "U \<sim> V" shows "freeleft U \<sim> freeleft V"
congdd_base_gen_congdd_base_aux: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and R' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "rel (cong.gen_cong rel eval R) x y"   and "R \<le> R'"   and "cong.cong rel eval R'" shows "R' (eval x) (eval y)"
simps: fixes a :: "'b formula" shows "AX10p (a::'b formula) = ((\<exists>F::'b formula. a = F \<and> AX0p F) \<or> (\<exists>(F::'b formula) G::'b formula. a = F \<^bold>\<rightarrow> F \<^bold>\<or> G) \<or> (\<exists>(G::'b formula) F::'b formula. a = G \<^bold>\<rightarrow> F \<^bold>\<or> G) \<or> (\<exists>(F::'b formula) (H::'b formula) G::'b formula. a = (F \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<or> G) \<^bold>\<rightarrow> H) \<or> (\<exists>(F::'b formula) G::'b formula. a = (F \<^bold>\<and> G) \<^bold>\<rightarrow> F) \<or> (\<exists>(F::'b formula) G::'b formula. a = (F \<^bold>\<and> G) \<^bold>\<rightarrow> G) \<or> (\<exists>(F::'b formula) G::'b formula. a = F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<^bold>\<and> G) \<or> (\<exists>F::'b formula. a = (F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> F) \<or> (\<exists>F::'b formula. a = \<^bold>\<not> F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom>) \<or> (\<exists>F::'b formula. a = (\<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F))"
inf_comm: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
add_existing_point: fixes p :: "'a" assumes "p \<in> \<V>\<^sup>+" shows "combine_sys.add_point p = \<V>\<^sup>+"
no_marker_and_snapshotted_implies_no_more_markers_trace: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "ps (S t i) p \<noteq> None"   and "Marker \<notin> set (msgs (S t i) cid)"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (S t j) cid)"
ocircline_set_of_circline: fixes H :: "circline" shows "ocircline_set (of_circline H) = circline_set H"
tendsto_at_topI_sequentially_real: fixes f :: "real \<Rightarrow> real"   and y :: "real" assumes "mono f"   and "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y" shows "(f \<longlongrightarrow> y) at_top"
nat_to_list_th: fixes ls :: "nat list" shows "nat_to_list (list_to_nat ls) = ls"
simps_2: fixes x :: "char list"   and a :: "aexp" shows "strip (x ::= a) = x ::= a"
iso_homology_contractible_space_subtopology2: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "S \<subseteq> topspace X"   and "p \<noteq> 0"   and "S \<noteq> {}" shows "homology_group p (subtopology X S) \<cong> relative_homology_group (p + 1) X S"
spanE: fixes p :: "'b"   and B :: "'b set" assumes "p \<in> local.span B" obtains A :: "'b set" and q :: "'b \<Rightarrow> 'a" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b *s b)"
tllist_all2_TCons1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and x :: "'g"   and ts :: "('g, 'i) tllist"   and ts' :: "('h, 'j) tllist" shows "tllist_all2 P Q (TCons x ts) ts' = (\<exists>x' ts''. ts' = TCons x' ts'' \<and> P x x' \<and> tllist_all2 P Q ts ts'')"
conv_radius_geI: fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" assumes "summable (\<lambda>n. f n * z ^ n)" shows "ereal (norm z) \<le> conv_radius f"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (Y a' f) = Cop.arr f"
cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "o[F](f) = o[F](g)"
t_op_def: fixes x :: "'a" shows "tx = at (at x)"
distinct_enum_upto': fixes a :: "'b word"   and b :: "'b word" shows "distinct [a .e. b]"
mono_det_D: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>D P'"   and "S \<sqsubseteq>\<^sub>D S'" shows "(P [+] S) \<sqsubseteq>\<^sub>D (P' [+] S')"
toSet_eq_iff: fixes X :: "'b odlist"   and Y :: "'b odlist" shows "(X = Y) = (toSet X = toSet Y)"
index_set_def: fixes I :: "nat set" shows "index_set I \<equiv> \<forall>i j. i \<in> I \<and> (\<forall>x. eval r_phi [i, x] = eval r_phi [j, x]) \<longrightarrow> j \<in> I"
MapsToI: fixes f :: "'a"   and C :: "('b, 'a, 'c) Category_scheme"   and X :: "'b"   and Y :: "'b" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "dom\<^bsub>C\<^esub> f = X"   and "cod\<^bsub>C\<^esub> f = Y" shows "f maps\<^bsub>C\<^esub> X to Y"
notin_List_update: fixes a :: "'c"   and as :: "'c list"   and h :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "a \<notin> set as" shows "List (h(a := y)) x as = List h x as"
1: fixes a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "a \<in> A"   and "a \<noteq> 0"   and "lt a adds\<^sub>t u" shows "rword (spp_of a) (spp_of p)"
correctCompositionDiffLevelsA91: shows "correctCompositionDiffLevels sA91"