hcju/isabelleps huggingface dataset:
nodes_trans: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and q :: "nat \<times> 'c" assumes "p \<in> gnodes A w V"   and "q \<in> greachable A w V p" shows "q \<in> gnodes A w V"
hyps_1: shows "cx \<turnstile> n -et\<rightarrow> (_Exit_)"
fref_def: fixes P :: "'i \<Rightarrow> bool"   and R :: "('g \<times> 'i) set"   and S :: "('h \<times> 'j) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
p: shows "is_path_undir (add_edge a w b G) v p v'"
shD: shows "sh D = \<lfloor>(sfs, i)\<rfloor>"
ins: fixes u :: "complex"   and z :: "complex" assumes "cmod u = 1"   and "z \<in> s" shows "u * z \<in> s"
relcompI: fixes a :: "'h"   and b :: "'i"   and r :: "('h \<times> 'i) set"   and c :: "'j"   and s :: "('i \<times> 'j) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
fls_integral_nth: fixes a :: "'a fls"   and n :: "int" shows "fls_nth (fls_integral a) n = (if n = 0 then 0::'a else inverse (of_int n) * fls_nth a (n - 1))"
nasc_seqn: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "nasc_seq A a n < nasc_seq A a (Suc n)"
0: shows "ns (k + Suc m) = ns (k + m) \<ominus> local.divide (f \<bullet> ns (k + m)) (f' \<bullet> ns (k + m))"
const_prop_def: fixes c :: "'b" shows "const_prop c \<equiv> transl_rule (A_Lbl (S_Const c) \<sqsubseteq> \<one>)"
nu_par_aux4: fixes x :: "'a"   and y :: "'a" shows "\<nu> (\<tau> x \<parallel> \<tau> y) = (0::'a)"
abc_Hoare_unhaltI: fixes P :: "nat list \<Rightarrow> bool"   and p :: "abc_inst list" assumes "\<And>args n. P args \<Longrightarrow> abc_notfinal (abc_steps_l (0, args) p n) p" shows "{P} p \<up>"
pointermap_p_valid_def: fixes p :: "nat"   and m :: "('b, 'c) pointermap_scheme" shows "pointermap_p_valid p m \<equiv> p < length (entries m)"
upd: shows "bij_betw u_t {..<n} {..<n}"
l_repr_independence: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <# H"   and "x \<in> carrier G"   and "subgroup H G" shows "x <# H = y <# H"
INF_lower: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and i :: "'c::type"   and A :: "'c::type set"   and f :: "'c::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(i::'c::type) \<in> (A::'c::type set)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set))) (f (i::'c::type))"
join_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<sqsubseteq> x \<squnion> y"
SafeStickyNrecVacantUpto_def: fixes s :: "('c, 'd) configuration stream" shows "SafeStickyNrecVacantUpto s = (let c = shd s in \<forall>t. NrecVacantUpto c t \<longrightarrow> alw (holds (\<lambda>c. NrecVacantUpto c t)) s)"
domain_eq: shows "Domainp pcr_rat = (\<lambda>x. x \<approx> x)"
args_are_strictly_lower: fixes t :: "'a trm" assumes "is_compound t" shows "(lhs t, t) \<in> trm_ord \<and> (rhs t, t) \<in> trm_ord"
assert_injective: fixes p :: "'b Assertion"   and q :: "'b Assertion" assumes "{\<cdot> p } = {\<cdot> q }" shows "p = q"
sat_transf_constrs_sat_constr: fixes cs :: "constraint"   and v :: "nat \<Rightarrow> rat" assumes "nonstrict_constr cs"   and "v \<Turnstile>\<^sub>c\<^sub>s set (transf_constraint cs)" shows "v \<Turnstile>\<^sub>c cs"
simps_1: fixes c :: "poly"   and n :: "nat"   and p :: "poly" shows "headconst (CN c n p) = headconst p"
traces_def: fixes A :: "('d, 'c) ioa" shows "local.traces A \<equiv> {tr. is_trace_of A tr}"
comp_inv_arr: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.inverse_arrows f g" shows "B.VV.comp g f = B.VV.dom f"
local: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and t :: "'b net_state"   and a :: "'c node_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "'b net_state"   and t' :: "'b net_state" assumes "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U"   and "((\<sigma>, SubnetS s t), a, \<sigma>', SubnetS s' t') \<in> automaton.trans (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2))"   and "S \<sigma> \<sigma>' a"   and "P \<sigma> s t" shows "P \<sigma>' s' t'"
assms_4: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
hs_in_sync_locs_def: shows "hs_in_sync_locs = - ((\<Union>l\<in>{''idle_noop'', ''idle_flip_noop'', ''init_noop'', ''mark_noop''} \<union> {''mark_loop_get_roots'', ''mark_loop_get_work''}. prefixed (l @ ''_init'')) \<union> hs_done_locs) \<union> (\<Union>l\<in>{''idle_noop'', ''idle_flip_noop'', ''init_noop'', ''mark_noop''} \<union> {''mark_loop_get_roots'', ''mark_loop_get_work''}. {l @ ''_init_type''})"
prems_1: shows "(\<Gamma>, e, S) \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> (\<Delta>, z, S)"
mult_nonneg_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
finite_enumerate_in_set: fixes S :: "'b set"   and n :: "nat" assumes "finite S"   and "n < card S" shows "wellorder_class.enumerate S n \<in> S"
Sup_mono: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq a b" shows "less_eq (Sup A) (Sup B)"
exec_append: fixes xs :: "('d, 'c) instr list"   and ys :: "('d, 'c) instr list"   and stack :: "'c list"   and env :: "'d \<Rightarrow> 'c" shows "exec (xs @ ys) stack env = exec ys (exec xs stack env) env"
aff_dim_convex_Int_nonempty_interior: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "S \<inter> interior T \<noteq> {}" shows "aff_dim (S \<inter> T) = aff_dim S"
gets_the_loop_terminates: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> 'b option"   and r :: "'b"   and s :: "'c" shows "whileLoop_terminates C (\<lambda>a. gets_the (B a)) r s = (\<exists>rs'. (Some r, rs') \<in> option_while' (\<lambda>a. C a s) (\<lambda>a. B a s))"
monotone_ldropn_aux: shows "monotone (\<sqsubseteq>) llist.le_fun (\<lambda>xs n. ldropn n xs)"
approx_mult_SComplex1: fixes a :: "complex star"   and x :: "complex star" assumes "a \<in> SComplex"   and "x \<approx> 0" shows "x * a \<approx> 0"
1: fixes A :: "('fun, 'var) Term.term set"   and B :: "('fun, 'var) Term.term set" assumes "tfr\<^sub>s\<^sub>e\<^sub>t (A \<union> B)" shows "tfr\<^sub>s\<^sub>e\<^sub>t A"
ide_diagonalize_can: fixes f :: "'c local.term set" assumes "\<F>C.can f" shows "\<F>C.ide (\<F>C.D f)"
axioms_3: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" assumes "linear s1 s2 f" shows "module_hom s1 s2 f"
Grp_fst_snd: fixes R :: "'n \<Rightarrow> 'o \<Rightarrow> bool" shows "(Grp {(x, y). R x y} fst)\<inverse>\<inverse> OO Grp {(x, y). R x y} snd = R"
bdd_all_bdd_map: fixes P :: "'a \<Rightarrow> bool"   and bdd :: "'a bdd"   and Q :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "bdd_all P bdd"   and "\<And>a. P a \<Longrightarrow> Q (f a)" shows "bdd_all Q (bdd_map f bdd)"
qiter_fun_def: fixes x :: "'b"   and y :: "'b" shows "qiter_fun x y = (\<squnion>) x \<circ> (\<cdot>) y"
l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_root_node get_owner_document \<equiv> get_owner_document = l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node get_disconnected_nodes"
sorted_sparse_row_vector_zero: fixes m :: "nat"   and n :: "nat"   and a :: "'a"   and arr :: "(nat \<times> 'a) list"   and j :: "nat" assumes "m \<le> n"   and "sorted_spvec ((n, a) # arr)" shows "Rep_matrix (sparse_row_vector arr) j m = (0::'a)"
has_derivative_imp_has_field_derivative: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'a"   and F :: "'a filter"   and D' :: "'a" assumes "(f has_derivative D) F"   and "\<And>x. x * D' = D x" shows "(f has_field_derivative D') F"
xspan: shows "lincomb l1 X \<in> local.span X"
HEndPhase0_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
obs_eq: shows "obs_intra (parent_node n) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs_intra (parent_node n') \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
componentsE: fixes S :: "'b set"   and U :: "'b set" assumes "S \<in> components U" obtains x :: "'b" where "x \<in> U" and "S = connected_component_set U x"
pathsCard_below: fixes ps :: "Vars.var list set"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>" assumes "\<And>p. p \<in> ps \<Longrightarrow> pathCard p \<sqsubseteq> ce" shows "pathsCard ps \<sqsubseteq> ce"
right_inverse: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "a * inverse a = (1::'a)"
semigroup_add_on_with_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "(rel_set A ===> (A ===> A ===> A) ===> (=)) semigroup_add_ow semigroup_add_ow"
contradicting_in_cp: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and \<sigma>' :: "'b \<Rightarrow> 'c"   and k :: "nat"   and k' :: "nat"   and v :: "'b" assumes "IFC prob"   and "IFC_def.loweq prob \<sigma> \<sigma>'"   and "IFC_def.cs prob (IFC_def.path prob \<sigma>) k = IFC_def.cs prob (IFC_def.path prob \<sigma>') k'"   and "v \<in> IFC_def.reads prob (IFC_def.path prob \<sigma> k)"   and "IFC_def.kth_state prob \<sigma> k v \<noteq> IFC_def.kth_state prob \<sigma>' k' v" shows "((\<sigma>, k), \<sigma>', k') \<in> IFC_def.cp prob"
simps_1: fixes r :: "real" shows "to_real_Gromov_completion (ereal r) = to_Gromov_completion r"
BseqE: fixes X :: "nat \<Rightarrow> 'a" assumes "Bseq X" obtains K :: "real" where "0 < K" and "\<forall>n. norm (X n) \<le> K"
LeftDerivationFix_derivation_ge: fixes \<alpha> :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list"   and d :: "nat" assumes "LeftDerivationFix \<alpha> i D j \<gamma>"   and "derivation_ge D d" shows "i = j \<or> d < i \<and> d \<le> j"
cyc_free_diag_dest': fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len m i i xs"   and "i \<le> n" shows "\<one> \<le> m i i"
less_numeral_extra_3: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
steph: fixes amount :: "nat"   and F :: "(atom list \<times> atom fm list \<times> nat) list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = amount \<and> eval (list_disj (map (\<lambda>(L, F, n). ExN n (list_conj (map fm.Atom L @ F))) F)) (xs @ \<Gamma>)) = eval (step amount F) \<Gamma>"
face_cycle_set_parts: fixes a :: "'b"   and b :: "'b" shows "H.face_cycle_set a = H.face_cycle_set b \<or> H.face_cycle_set a \<inter> H.face_cycle_set b = {}"
runit'_simps_4: fixes f :: "'d" assumes "D.ide f" shows "D.dom \<r>\<^sub>D\<^sup>-\<^sup>1[f] = f"
in3': shows "dim_row (two_block_non_interfering A B) - n + i < dim_row (two_block_non_interfering A B)"
that: fixes as' :: "'edge list" assumes "n -as'\<rightarrow>\<^sub>\<surd>* (_Exit_)"   and "set (sourcenodes as') \<subseteq> set (sourcenodes as)"   and "\<forall>a'\<in>set as'. intra_kind (kind a') \<or> (\<exists>Q f p. kind a' = Q\<hookleftarrow>\<^bsub>p\<^esub>f)" shows "thesis"
dbm_lt_asymmetric: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y"   and "y \<prec> x" shows "False"
intro: fixes set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')" shows "l_set_tag_name_get_child_nodes_axioms set_tag_name_locs get_child_nodes_locs"
l2_5_1_A: fixes S :: "('b \<times> 'b \<Rightarrow> bool) set" shows "\<forall>X\<in>S. \<forall>A. (X \<longrightarrow> (\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> S})) A"
proj2_incident_unique: fixes p :: "proj2"   and l :: "proj2_line"   and q :: "proj2"   and m :: "proj2_line" assumes "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident p m"   and "proj2_incident q m" shows "p = q \<or> l = m"
local_order_prefix_closed: fixes x :: "'evt"   and i :: "nat"   and y :: "'evt"   and xs :: "'evt list" assumes "x \<sqsubset>\<^sup>i y"   and "xs prefix of i"   and "y \<in> set xs" shows "x \<in> set xs"
times_eqpoll_cong: fixes A :: "'a set"   and C :: "'b set"   and B :: "'c set"   and D :: "'d set" assumes "A \<approx> C"   and "B \<approx> D" shows "A \<times> B \<approx> C \<times> D"
bij: shows "bij_betw (\<lambda>r. r * a mod k) {0..k - 1} {0..k - 1}"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> ideal S"
dghm_HomDom: shows "\<GG>\<lparr>HomDom\<rparr> = \<AA>"
frechet_derivative_transform_within_open_ext: fixes f :: "'b \<Rightarrow> 'c"   and xa :: "'b"   and X :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f differentiable at xa"   and "open X"   and "xa \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x" shows "frechet_derivative f (at xa) x = frechet_derivative g (at xa) x"
A2: shows "weak_reduction_contrasimulation (Rel\<inverse>) Cal"
the_wall_betw_nempty_2: fixes C :: "'a set"   and D :: "'a set" assumes "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C D) \<noteq> {}" shows "separated_by (THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C D)) C D"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {..<Suc n} = g 0 + (\<Sum>i<n. g (Suc i))"
d_JNF: shows "(P', S', Q') = diagonalize_JNF A' bezout"
a1: shows "low_equal s1 s2 \<and> \<not> snd (execute_instr_sub1 instr s1) \<and> \<not> snd (execute_instr_sub1 instr s2) \<and> t1 = snd (fst (execute_instr_sub1 instr s1)) \<and> t2 = snd (fst (execute_instr_sub1 instr s2))"
pair_spmf_return_spmf1: fixes x :: "'c"   and q :: "'d spmf" shows "pair_spmf (return_spmf x) q = map_spmf (Pair x) q"
vwalk_length_simp: fixes p :: "'a list" shows "vwalk_length p = length p - 1"
chine_hcomp_ide_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "span_vertical_category.src C f = span_vertical_category.trg C \<mu>" shows "span_bicategory.chine_hcomp C prj0 prj1 f \<mu> = elementary_category_with_pullbacks.tuple C prj0 prj1 (prj1 (Leg0 (Dom f)) (Leg1 (Dom \<mu>))) (Leg0 (Cod f)) (Leg1 (Cod \<mu>)) (C (Chn \<mu>) (prj0 (Leg0 (Dom f)) (Leg1 (Dom \<mu>))))"
intros_5: fixes d :: "'d"   and u :: "'e \<Rightarrow> 'd"   and c :: "'e" assumes "d \<le> u c" shows "u \<turnstile> GE c d"
bla: shows "\<forall>y. P y (\<sigma>x. F x) \<longrightarrow> (\<exists>z. F z \<and> O y z)"
zmultiset_eq_iff: fixes M :: "'a zmultiset"   and N :: "'a zmultiset" shows "(M = N) = (\<forall>a. zcount M a = zcount N a)"
equivalence_implies_bisimilarity: fixes P :: "'state"   and Q :: "'state" assumes "P =\<cdot> Q" shows "P \<sim>\<cdot> Q"
smap_sfilter_split_cfcomp: fixes f :: "'s \<rightarrow> 'r"   and p :: "'t \<rightarrow> tr"   and g :: "'s \<rightarrow> 't" assumes "f\<cdot>\<bottom> = \<bottom>"   and "p\<cdot>\<bottom> = \<bottom>" shows "smap\<cdot>f oo sfilter\<cdot>(p oo g) = smap\<cdot>cfst oo sfilter\<cdot>(p oo csnd) oo smap\<cdot>(f && g)"
emeasure_mono_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "AE x\<in>A in M. x \<in> B"   and "B \<in> sets M" shows "emeasure M A \<le> emeasure M B"
nwprops: shows "qty_rem_comp pf1 x y n w \<noteq> 0"
assms_2: shows "wa_refine WB WA \<alpha>2"
Fx_equiv_xH: fixes v :: "i" assumes "\<And>G \<phi>. IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)" shows "[\<^bold>\<forall>H. \<^bold>\<exists>F. \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lbrace>x\<^sup>P,H\<rbrace>) in v]"
int_numeral_bitOR_num: fixes n :: "num"   and m :: "num" shows "numeral n OR numeral m = numeral (bitOR_num n m)"
fm: shows "x = \<lbrakk>\<guillemotleft>fm\<guillemotright>\<rbrakk>e"
inverse_eq_divide: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse a = divide one a"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "C.ide (F a)"
exhaustiveUnitPropagatePreservedVariables: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = exhaustiveUnitPropagate state in getSATFlag state' = getSATFlag state"
exp_less_cancel_iff: fixes x :: "real"   and y :: "real" shows "(exp x < exp y) = (x < y)"
valid_lists_chain2: fixes Y :: "nat \<Rightarrow> CoCalls"   and xs :: "Vars.var list"   and S :: "Vars.var set" assumes "chain Y"   and "xs \<in> valid_lists S (\<Squnion>i. Y i)" shows "\<exists>i. xs \<in> valid_lists S (Y i)"
implConfid_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> valid \<union> range LtK \<union> Tags"   and "implConfid A B M \<in> synth (analz H)" shows "implConfid A B M \<in> H \<or> M \<in> synth (analz H)"
k_invariant_step_old: fixes x :: "'rep"   and ec :: "'rep"   and A :: "('ma, 'mt) AlgState"   and obs :: "'obs" assumes "k_isNode x"   and "k_isNode ec"   and "simAbs ec \<noteq> simAbs x" shows "lookup tOps (aTrans (k_ins x A)) (ec, obs) = lookup tOps (aTrans A) (ec, obs)"
case_transfer: fixes R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "((R2 ===> S) ===> rel_optionT' R1 R2 ===> S) case_optionT case_optionT"
c_tl_c_drop: fixes y :: "nat"   and x :: "nat" shows "c_tl (c_drop y x) = c_drop y (c_tl x)"
infinite_path_is_maximal: fixes P :: "'a llist" assumes "G'.valid_path P"   and "\<not> lfinite P" shows "G'.maximal_path P"
OrdNotEqP_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (t NEQ u)"
d_IN_Inf: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow f)"   and "\<And>g e. g \<in> Y \<Longrightarrow> g e \<le> f e"   and "d_IN f x \<noteq> \<top>" shows "d_IN (\<Sqinter> Y) x = (\<Sqinter>g\<in>Y. d_IN g x)"
Joints_msum_aform: fixes X :: "'b \<times> 'b pdevs"   and d :: "nat"   and YS :: "('b \<times> 'b pdevs) list"   and Y :: "'b \<times> 'b pdevs" assumes "degree_aform X \<le> d"   and "\<And>Y. Y \<in> set YS \<Longrightarrow> degree_aform Y \<le> d" shows "Joints (msum_aform d X Y # YS) = {(x + y) # ys |x y ys. y \<in> Affine Y \<and> x # ys \<in> Joints (X # YS)}"
is_sup_upper: fixes x :: "'c"   and y :: "'c"   and sup :: "'c" assumes "is_sup x y sup" obtains "x \<sqsubseteq> sup" and "y \<sqsubseteq> sup"
u_simps_2: shows "arr (tab\<^sub>0 r)"
elims_2: fixes x :: "sheap"   and xa :: "'a exp list" assumes "iconfs x xa" obtains sh :: "sheap" where "x = sh__" and "xa = []"   | sh :: "sheap" and e :: "'a exp" and es :: "'a exp list" where "x = sh" and "xa = e # es" and "case val_of e of None \<Rightarrow> iconf sh e \<and> \<not> sub_RIs es | \<lfloor>v\<rfloor> \<Rightarrow> iconfs sh es"
intros_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "list_emb P [] ys"
all_after_eq_7: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<exists>x. P x)$ \<equiv> \<exists>x. P x$"
g12: fixes x :: "'a" assumes "x \<in> sphere (0::'a) 1 \<inter> S" shows "norm (f x - g x) < 1 / 2"
edge'_simps_6: fixes xo :: "'v vertex" shows "edge' xo SOURCE = False"