hcju/isabelleps huggingface dataset:
of_pred_code: fixes f :: "unit \<Rightarrow> 'b Predicate.seq" shows "set_of_pred (Predicate.Seq f) = (case f () of seq.Empty \<Rightarrow> {} | seq.Insert x P \<Rightarrow> insert x (set_of_pred P) | seq.Join P xq \<Rightarrow> set_of_pred P \<union> set_of_seq xq)"
insert_rga_Some_commutes: fixes i1 :: "'a"   and xs :: "'a list"   and i2 :: "'a"   and e1 :: "'a"   and e2 :: "'a" assumes "i1 \<in> set xs"   and "i2 \<in> set xs"   and "e1 \<noteq> i2"   and "e2 \<noteq> i1" shows "insert_rga (insert_rga xs (e1, Some i1)) (e2, Some i2) = insert_rga (insert_rga xs (e2, Some i2)) (e1, Some i1)"
assms: shows "f \<in> borel_measurable (stream_space M)"
nofail_SPEC_triv_refine: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "nofail m"   and "\<And>x. \<Phi> x" shows "m \<le> SPEC \<Phi>"
gcd_lcm: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "gcd a b = normalize (divide (times a b) (lcm a b))"
respectfulness_6: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "step \<phi> \<nu> \<equiv>\<^sub>P step \<psi> \<nu>"
ring_hom_minus: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and a :: "'c"   and b :: "'c" assumes "ring R"   and "ring S"   and "h \<in> ring_hom R S"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "h (a \<ominus>\<^bsub>R\<^esub> b) = h a \<ominus>\<^bsub>S\<^esub> h b"
special_coset_below_in_supset_ordering_iso: fixes T :: "'w set"   and w :: "'w" shows "OrderingSetIso order.greater_eq order.greater order.greater_eq order.greater (dual_order.below_in (Pow S) T) ((+o) w \<circ> genby)"
has_complex_derivative_locally_injective: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "\<xi> \<in> S"   and "open S"   and "deriv f \<xi> \<noteq> 0" obtains r :: "real" where "0 < r" and "ball \<xi> r \<subseteq> S" and "inj_on f (ball \<xi> r)"
apply_wiring_simps: fixes f :: "'k \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'l" shows "apply_wiring (f, g) = map_fun id (map_fun f (map_spmf (map_prod g id)))"
symmetric_point_construction: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<exists>P'. Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"
alw_holds_sset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "alw (holds P) xs = (\<forall>x\<in>sset xs. P x)"
lin_part_def: fixes f :: "nat \<Rightarrow> 'a" shows "lin_part f = trms_of_deg_leq 1 f"
closed_orbit_flow_compact: fixes x :: "'a" assumes "closed_orbit x" shows "compact (range (flow0 x))"
insert_extract_weights_cong_unshared: fixes m :: "real mat convnet"   and f :: "nat \<Rightarrow> real" assumes "\<And>x. x < count_weights False (remove_weights m) \<Longrightarrow> f x = extract_weights False m x" shows "m = insert_weights False (remove_weights m) f"
assms: shows "i < card {i. i < dim_row A \<and> i \<in> I}"
all: fixes pa :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "pa \<in> \<P> k u" shows "P pa"
cond_L6: fixes b :: "'c upred"   and P :: "('b, 'c) uexpr"   and Q :: "('b, 'c) uexpr"   and R :: "('b, 'c) uexpr" shows "P \<triangleleft> b \<triangleright> (Q \<triangleleft> b \<triangleright> R) = P \<triangleleft> b \<triangleright> R"
Dugundji: fixes C :: "'a set"   and U :: "'b set"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "convex C"   and "C \<noteq> {}"   and "closedin (top_of_set U) S"   and "continuous_on S f"   and "f ` S \<subseteq> C" obtains g :: "'b \<Rightarrow> 'a" where "continuous_on U g" and "g ` U \<subseteq> C" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
assms_3: fixes w :: "(real, 'n) vec" assumes "w \<in> S" shows "orthogonal v w"
complements_do_not_falsify': fixes l\<^sub>1 :: "fterm literal"   and C\<^sub>1' :: "fterm literal set"   and l\<^sub>2 :: "fterm literal"   and G :: "bool list" assumes "l\<^sub>1 \<in> C\<^sub>1'"   and "l\<^sub>2 \<in> C\<^sub>1'"   and "l\<^sub>1 = l\<^sub>2\<^sup>c"   and "falsifies\<^sub>g G C\<^sub>1'" shows "False"
p_subid_lb1: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<parallel> (y \<cdot> (0::'a)) \<le> x \<cdot> (0::'a)"
Quotient3_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c" shows "Quotient3 R Abs Rep = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)))"
psi_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "psi m \<le> psi n"
103: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> - y = x"
a_r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H +> x \<subseteq> carrier G"
snoc_3: shows "\<S>\<bullet>\<C> \<turnstile> ves @ [a] : [] _> tvs"
Range_converse: fixes r :: "('a \<times> 'b) set" shows "Range (r\<inverse>) = Domain r"
orthogonal_matrix_is_orthogonal: fixes A :: "((real, 'b) vec, 'b) vec" assumes "orthogonal_matrix A" shows "pairwise orthogonal (columns A)"
divide_less_eq_1_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less (divide b a) one = less b a"
indep_in_imp_subset_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.indep_in \<E> X" shows "\<exists>B. \<E>.basis_in \<E> B \<and> X \<subseteq> B"
rel_pred_pos_distr_cond_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pred_pos_distr_cond A B = (rel_pred A OO rel_pred B \<le> rel_pred (A OO B))"
simps_5: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d = (\<lambda>\<I>. ineq_model \<I> X F \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d \<I>)"
steps: shows "(s, t) \<in> NS\<^sup>* O S"
iso_points_inv_map: shows "inv_into \<V> \<pi> ` \<V>' = \<V>"
open_closed_segment: fixes u :: "'c"   and w :: "'c"   and z :: "'c" assumes "u \<in> {w<--<z}" shows "u \<in> {w--z}"
simps: fixes a1 :: "'d + 'd"   and a2 :: "'d + 'd" shows "ignore_Inl a1 a2 = ((\<exists>x y. a1 = Inl x \<and> a2 = Inl y) \<or> (\<exists>x. a1 = Inr x \<and> a2 = Inr x))"
fdoubleton_eq_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({|a, b|} = {|c, d|}) = (a = c \<and> b = d \<or> a = d \<and> b = c)"
length_reduct: fixes m :: "int poly fract list list" assumes "m \<noteq> []" shows "length (rat_poly.reduct m) + 1 = length m"
le_iff_sup: fixes sup :: "'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 x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_sup (sup::'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)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type) = ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x y = y)"
normal_poly_pos_interval: fixes p :: "'a poly"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "normal_poly p"   and "i \<le> j"   and "j \<le> k"   and "(0::'a) < coeff p i"   and "(0::'a) < coeff p k" shows "(0::'a) < coeff p j"
alist_of_inject: fixes x :: "('c, 'd) pq"   and y :: "('c, 'd) pq" shows "(pq.alist_of x = pq.alist_of y) = (x = y)"
kcomp_distr: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'a \<Rightarrow> 'c set"   and h :: "'c \<Rightarrow> 'b set" shows "(f \<squnion> g) \<circ>\<^sub>K h = f \<circ>\<^sub>K h \<squnion> g \<circ>\<^sub>K h"
mndet_chainpreserving: fixes Y :: "nat \<Rightarrow> 'a \<Rightarrow> 'a process"   and A :: "'a set" assumes "chain Y" shows "chain (\<lambda>i. mndet A (Y i))"
at_within_nhd: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and S :: "'b set"   and T :: "'b set"   and U :: "'b set" assumes "class.topological_space open"   and "x \<in> S"   and "open S"   and "T \<inter> S - {x} = U \<inter> S - {x}" shows "topological_space.at_within open x T = topological_space.at_within open x U"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_begin0 n (l, r) = (1 < n \<and> (l, r) = (Oc \<up> (n - 2), [Oc, Oc, Bk, Oc]) \<or> n = 1 \<and> (l, r) = ([], [Bk, Oc, Bk, Oc]))"
simps_1: fixes x :: "'b" shows "one_call_in_path x [] = True"
Inr_not_in_new_spairs: fixes j :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "Inr j \<notin> set (new_spairs bs p)"
to_nat_on_inj: fixes A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "countable A"   and "a \<in> A"   and "b \<in> A" shows "(to_nat_on A a = to_nat_on A b) = (a = b)"
quotientE: fixes X :: "'c set"   and A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "X \<in> A // r" obtains x :: "'c" where "X = r `` {x}" and "x \<in> A"
\<ll>'_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<ll>' a = \<l>\<^sup>-\<^sup>1[a]"
simps_12: fixes p :: "nat"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p Pi vs = Some (ivl_err (real_interval (pi_float_interval p)))"
l_sort_def: fixes xs :: "'b list" shows "l_sort xs \<equiv> l_sort_out (l_sort_aux (l_sort_in xs))"
terminates_Cons_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and x :: "'f" assumes "terminates g" shows "terminates (Cons_raw x g)"
assms_3: shows "psi n \<le> 3 / 2 * ln 2 * real n"
trnl\<^sub>\<eta>_def: fixes v :: "'a"   and \<mu> :: "'a" shows "T0.trnl\<^sub>\<eta> v \<mu> \<equiv> (f\<^sup>* \<star> \<mu>) \<cdot> \<a>[f\<^sup>*, f, v] \<cdot> (\<eta> \<star> v) \<cdot> \<l>\<^sup>-\<^sup>1[v]"
plus_le_left_cancel_wrap: fixes x :: "'b word"   and y' :: "'b word"   and y :: "'b word" assumes "x + y' < x"   and "x + y < x" shows "(x + y' < x + y) = (y' < y)"
cond: fixes i :: "nat" assumes "i < len" shows "cond (Is i) (Is (Suc i)) (ls i) (cs i) (hs i) (ks i)"
simps_6: fixes \<alpha> :: "('a, 'b, 'c) hp"   and \<phi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (\<langle> \<alpha> \<rangle> \<phi>) \<sigma> = (\<langle> Psubst \<alpha> \<sigma> \<rangle> Fsubst \<phi> \<sigma>)"
distinct_103: fixes x7 :: "logic_type"   and x8 :: "shift_type" shows "logic_type x7 \<noteq> shift_type x8"
composites_are_arrows_2: shows "hseq (\<mu> \<star> \<nu>) \<pi>"
simps_5: fixes x2 :: "'b"   and x1 :: "'a" shows "In2 x2 \<noteq> In1 x1"
bij_betw_arr_sets: shows "bij_betw \<psi>.inv (Collect J'J.A.arr) (Collect J.arr)"
is_unit_pCons_iff: fixes a :: "'c"   and p :: "'c poly" shows "is_unit (pCons a p) = (p = 0 \<and> a \<noteq> (0::'c))"
star_of_inverse: fixes x :: "'c" shows "star_of (inverse x) = inverse (star_of x)"
min_absorb1: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "min x y = x"
assms: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> edges g"   and "(u, v) \<in> (edges g - {(u, v), (v, u)})\<^sup>*" shows "False"
simps_1: fixes p :: "fm"   and q :: "fm" shows "iszlfm (And p q) = (iszlfm p \<and> iszlfm q)"
outstanding_refs_is_Write\<^sub>s\<^sub>b_takeWhile_disj: shows "\<forall>i<length ts\<^sub>s\<^sub>b. \<forall>j<length ts\<^sub>s\<^sub>b. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>i, uu_, uu_, uu_) = ts\<^sub>s\<^sub>b ! i; (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts\<^sub>s\<^sub>b ! j in outstanding_refs is_Write\<^sub>s\<^sub>b sb\<^sub>i \<inter> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) = {})"
le_add_diff: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "c \<subseteq># b + c - a"
emeasure_measurable_subprob_algebra2: fixes M :: "'c measure"   and A :: "'c \<Rightarrow> 'd set"   and N :: "'d measure"   and L :: "'c \<Rightarrow> 'd measure" assumes "Sigma (space M) A \<in> sets (M \<Otimes>\<^sub>M N)"   and "L \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "(\<lambda>x. emeasure (L x) (A x)) \<in> borel_measurable M"
in_hhomE: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and a :: "'c"   and b :: "'c" assumes "horizontal_homs V src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b" obtains "partial_magma.arr V \<mu>" and "horizontal_homs.obj V src a" and "horizontal_homs.obj V src b" and "src \<mu> = a" and "trg \<mu> = b"
assms_10: shows "selected_edge h j g \<le> - forest_components f"
simps: fixes s :: "name"   and v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and k :: "name" assumes "atom s \<sharp> (v, i, t, u, k)"   and "atom k \<sharp> (v, i, t, u)" shows "SubstTermP v i t u = TermP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i t u (Var s) (Var k)))"
list_max_maximum: fixes a :: "'t"   and xs :: "'t list" assumes "a \<in> set xs" shows "a \<preceq>\<^sub>t list_max xs"
edge_antiparallel: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma>' x y" shows "\<not> edge \<Gamma>' y x"
urel64_inverse: fixes pp :: "uint64"   and x :: "uint64"   and x' :: "int" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295"   and "urel64 x x'" shows "urel64 (inverse_p64 pp x) (inverse_p p x')"
simps: fixes p :: "real mpoly"   and var :: "nat" shows "allZero p var = list_conj (map (\<lambda>i. fm.Atom (Eq (isolate_variable_sparse p var i))) [0..<MPoly_Type.degree p var + 1])"
aux_lemma: fixes pa :: "'a list"   and qa :: "'a list" assumes "pa \<in> carrier (K [X])"   and "qa \<in> carrier (K [X])"   and "pa \<sim>\<^bsub>K [X]\<^esub> qa" shows "length pa \<le> length qa"
ceq: fixes ceq :: "('c \<Rightarrow> 'c \<Rightarrow> bool) option"   and eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.ceq ceq"   and "ceq = Some eq" shows "eq = (=)"
right_uniqueD: fixes A :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'b"   and y :: "'e"   and z :: "'e" assumes "right_unique A"   and "A x y"   and "A x z" shows "y = z"
state_to_strips_state_dom_is_i: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'e option" assumes "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). v \<in> dom (range_of \<Psi>)" shows "set (concat (map (possible_assignments_for \<Psi>) (filter (\<lambda>v. s v \<noteq> None) (\<Psi>\<^sub>\<V>\<^sub>+)))) = (\<Union>v\<in>{v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None}. {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
left_adjoint_extends_to_adjunction: 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"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "chosen_right_adjoints V H \<a> \<i> src trg"   and "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g" shows "adjunction_in_bicategory V H \<a> \<i> src trg f (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f) (chosen_right_adjoints.some_unit V H \<a> \<i> src trg f) (chosen_right_adjoints.some_counit V H \<a> \<i> src trg f)"
assms_2: shows "\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
emb_step_at_right: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at [] dir.Right (App t s) = s"
qfree_qe_FMo\<^sub>1: fixes as :: "atom list" shows "qfree (qe_FMo\<^sub>1 as)"
ref_two_step: fixes A :: "'f nres"   and R :: "('f \<times> 'g) set"   and B :: "'g nres"   and C :: "'g nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>P, \<alpha>\<cdot>(\<tau>.Q)) \<in> Rel"
characteristic_iso_4: fixes f :: "'a" assumes "S.ide f" shows "S.iso ((\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f])"
Join_SKIP_left: fixes F :: "'c program" shows "\<bottom> \<squnion> F = F"
nat_0_1_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P 1"   and "\<And>n. \<lbrakk>1 \<le> n; P n\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P n"
mtf_def: fixes x :: "'b"   and xs :: "'b list" shows "mtf x xs = (if x \<in> set xs then x # take (index xs x) xs @ drop (index xs x + 1) xs else xs)"
relcomppI: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'d"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "r a b"   and "s b c" shows "(r OO s) a c"
results'_gpv_catch_gpv: fixes gpv :: "('d, 'e, 'f) gpv" shows "results'_gpv (catch_gpv gpv) = Some ` results'_gpv gpv \<union> (if colossless_gpv \<I>_full gpv then {} else {None})"
elims: fixes x :: "('c, 'd) stateful_strand_step list"   and xa :: "'d \<Rightarrow> ('c, 'd) Term.term"   and xb :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set"   and y :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" assumes "dbupd\<^sub>s\<^sub>s\<^sub>t x xa xb = y" obtains I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = []" and "xa = I__" and "xb = D__" and "y = D__"   | t :: "('c, 'd) Term.term" and s :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = insert\<langle>ta__,s__\<rangle> # A__" and "xa = Ia__" and "xb = Da__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t A__ Ia__ (insert ((ta__, s__) \<cdot>\<^sub>p Ia__) Da__)"   | t :: "('c, 'd) Term.term" and s :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = delete\<langle>t,s\<rangle> # Aa__" and "xa = Ib__" and "xb = Db__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t Aa__ Ib__ (Db__ - {(t, s) \<cdot>\<^sub>p Ib__})"   | v :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = send\<langle>v__\<rangle> # Ab__" and "xa = Ic__" and "xb = Dc__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t Ab__ Ic__ Dc__"   | v :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = receive\<langle>va__\<rangle> # Ac__" and "xa = Id__" and "xb = Dd__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t Ac__ Id__ Dd__"   | v :: "poscheckvariant" and va :: "('c, 'd) Term.term" and vb :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = \<langle>vb__: vaa__ \<doteq> vba__\<rangle> # Ad__" and "xa = Ie__" and "xb = De__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t Ad__ Ie__ De__"   | v :: "poscheckvariant" and va :: "('c, 'd) Term.term" and vb :: "('c, 'd) Term.term" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = \<langle>vc__: vab__ \<in> vbb__\<rangle> # Ae__" and "xa = If__" and "xb = Df__" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t Ae__ If__ Df__"   | v :: "'d list" and va :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" and vb :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" and A :: "('c, 'd) stateful_strand_step list" and I :: "'d \<Rightarrow> ('c, 'd) Term.term" and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" where "x = \<forall>v\<langle>\<or>\<noteq>: va \<or>\<notin>: vb\<rangle> # A" and "xa = I" and "xb = D" and "y = dbupd\<^sub>s\<^sub>s\<^sub>t A I D"
U_consts_2: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> U\<^sub>n false\<^sub>n"
poly_mapping_size_0: shows "local.poly_mapping_size 0 = g (0::'b)"
bounded_clinear_one_dim_iso: shows "bounded_clinear one_dim_iso"
elims: fixes x :: "trm"   and xa :: "stack"   and y :: "trm" assumes "x \<star> xa = y"   and "\<And>t. \<lbrakk>x = t; xa = stack.Id; y = t\<rbrakk> \<Longrightarrow> thesis"   and "\<And>xb K t s. \<lbrakk>xb \<sharp> (K, t); x = t; xa = [xb]s\<ggreater>K\<rbrakk> \<Longrightarrow> True" obtains xb :: "name" and K :: "stack" and t :: "trm" and s :: "trm" where "xb \<sharp> (K, t)" and "x = t" and "xa = [xb]s\<ggreater>K" and "y = (t to xb in s) \<star> K"
alphaAbs_qAbs_iff_alphaAbs_ex_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_qFresh xs x X xs' x' X'"
lsu_rpc: fixes x :: "'b"   and y :: "'b" shows "\<nu>\<^sup>\<natural> (rpc x y) = rpc x y"
unit_factor_poly_def: fixes p :: "'b poly" shows "unit_factor p = [:unit_factor (lead_coeff p):]"
v_S1: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "state cfg \<in> S1" shows "v cfg = \<integral>\<^sup>+ s. v (cont cfg s) \<partial>measure_pmf (action cfg)"
ln_shifted_float: fixes m :: "int"   and e :: "int" assumes "0 < m" shows "ln (real_of_float (Float m e)) = ln 2 * real_of_int (e + (bitlen m - 1)) + ln (real_of_float (Float m (- (bitlen m - 1))))"
unifiable_insert_swap: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term"   and E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set" shows "unifiable (insert (s, t) E) = unifiable (insert (t, s) E)"
xyz_1: shows "(X, Y) \<in> mult2_alt b1 ns s"
integrable_minus: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" shows "integrable M (\<lambda>x. - f x)"
assms: shows "sum_list (map (num_funs \<circ> (\<lambda>t. t \<cdot> \<sigma>)) ts) = sum_list (map num_funs ts)"
simps_1: fixes l2 :: "'b list" shows "inter_sorted_rev [] l2 = []"
\<II>_is_generated_by_\<PP>: shows "\<II> = Gen \<PP>"
set_shadow_root_writes: fixes element_ptr :: "(_) element_ptr"   and tag :: "(_) shadow_root_ptr option"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr tag) h h'"
lim_const_over_n: fixes a :: "'c" shows "(\<lambda>n. a / of_nat n) \<longlonglongrightarrow> (0::'c)"