hcju/isabelleps huggingface dataset:
bind_commute_spmf: fixes p :: "'c spmf"   and q :: "'d spmf"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
dsum_bound: fixes p :: "'a set set"   and a :: "'a"   and b :: "'a"   and c :: "'b"   and e :: "real" assumes "p division_of cbox a b"   and "norm c \<le> e" shows "norm (\<Sum>l\<in>p. content l *\<^sub>R c) \<le> e * content (cbox a b)"
below1: shows "\<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace> (x, e2) # \<Gamma> \<rbrace>\<rho>\<^esub>"
ifbd_ifbb_demorgan: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "bd\<^sup>-\<^sub>\<F> \<phi> = (bb\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
\<R>_closed_glob_consistent: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma>' :: "'a \<Rightarrow> 'd set option"   and \<S>' :: "'a set \<times> 'a set"   and P' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_security_init.closed_glob_consistent dma \<C>_vars (sifum_types.\<R> ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma>' \<S>' P')"
atU_single: fixes U :: "'f set"   and s :: "'f list" shows "atU U {#s#} = atU_s U s"
arcsin_le_mono: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> \<le> 1" shows "(arcsin x \<le> arcsin y) = (x \<le> y)"
Pmono: shows "\<forall>nP e1 e2 N E' E. wf_graph \<lparr>nodes = N, edges = E\<rparr> \<and> (e1, e2) \<in> E \<and> E' \<subseteq> E \<and> P e1 e2 nP \<lparr>nodes = N, edges = E\<rparr> \<longrightarrow> P e1 e2 nP \<lparr>nodes = N, edges = E'\<rparr>"
c2: shows "1 / det B \<cdot>\<^sub>m adj_mat B \<in> carrier_mat n n"
BVG_ODE: fixes x :: "char"   and \<theta> :: "trm" shows "BVG (ODE x \<theta>) \<subseteq> {RVar x, DVar x}"
d1_2: fixes xa :: "'a" assumes "dist xa x \<le> d1" shows "\<bar>u xa\<bar> < e"
sorted_sorted_wrt: shows "ord_term_lin.sorted = sorted_wrt (\<preceq>\<^sub>t)"
measurement_res_def: fixes M :: "complex mat"   and \<rho> :: "complex mat" shows "measurement_res M \<rho> = M * \<rho> * adjoint M"
filter_M_is_OK_I: fixes xs :: "'a list"   and h :: "'b"   and P :: "'a \<Rightarrow> ('b, 'c, bool) prog" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok P x"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> ok filter_M P xs"
values_reduce_to_themselves: fixes v :: "trm" assumes "val v" shows "v \<Down> v"
ide_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "ide a = \<guillemotleft>a : a \<rightarrow> a\<guillemotright>"
C': shows "\<forall>w\<in>set (cols A). dim_vec w = n"
nat_pow_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<in> carrier G"
utos_def: fixes w :: "bit list" shows "utos w = norm_signed (\<zero> # w)"
exact_seq_imp_group_hom_arbitrary: fixes G :: "'b monoid list"   and F :: "('b \<Rightarrow> 'b) list"   and i :: "nat" assumes "exact_seq (G, F)"   and "i < length F" shows "group_hom (G ! Suc i) (G ! i) (F ! i)"
NatTransExt: fixes \<eta>1 :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and \<eta>2 :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTrans \<eta>1"   and "NatTrans \<eta>2"   and "NTDom \<eta>1 = NTDom \<eta>2"   and "NTCod \<eta>1 = NTCod \<eta>2"   and "\<And>X. X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub> \<Longrightarrow> \<eta>1 $$ X = \<eta>2 $$ X" shows "\<eta>1 = \<eta>2"
f_i_def: shows "f_i = rec_nat (\<lambda>_. 0) (\<lambda>i f_i. let g = f_aux (Suc i) \<ominus> f_i in Let (SOME k. k \<le> g \<and> flow (residual_network f_i) k \<and> value_flow (residual_network f_i) k = value_flow (residual_network f_i) g \<and> (\<forall>x. d_IN k x \<le> value_flow (residual_network f_i) k)) ((\<oplus>) f_i))"
continuous_open_quotient_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "open_map X X' f" shows "quotient_map X X' f = (f ` topspace X = topspace X')"
boundedGreatest: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat"   and n :: "nat" assumes "P i"   and "\<forall>n'>n. \<not> P n'" shows "\<exists>i'\<le>n. P i' \<and> (\<forall>n'. P n' \<longrightarrow> n' \<le> i')"
count_wc_queries_aux: fixes Rs :: "('a \<times> 'a) set set"   and sorter :: "'a sorter"   and sorter' :: "('a \<times> 'a) set \<Rightarrow> 'a sorter"   and Rs' :: "('a \<times> 'a) set set" assumes "\<And>R. R \<in> Rs \<Longrightarrow> sorter = sorter' R"   and "Rs \<subseteq> Rs'"   and "finite Rs'" shows "count_wc_queries Rs sorter \<le> (MAX R\<in>Rs'. count_queries R (sorter' R))"
compute_lapprox_posrat: fixes prec :: "nat"   and x :: "nat"   and y :: "nat" shows "lapprox_posrat prec x y = (let l = rat_precision prec (int x) (int y); x = if 0 \<le> l then x * 2 ^ nat l div y else x div 2 ^ nat (- l) div y in normfloat (Float (int x) (- l)))"
simps_15: fixes x1 :: "'b" shows "funs_term (Var x1) = {}"
out_subseqs2_set_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and tail :: "'d iarray"   and v :: "'c"   and js :: "nat list" shows "my_subseqs.out_subseqs2_set f tail v js \<equiv> {f (tail !! j) v |j. j \<in> set js}"
subalgebra_def: fixes M :: "'a measure"   and F :: "'a measure" shows "subalgebra M F = (space F = space M \<and> sets F \<subseteq> sets M)"
r'': shows "Seq r'' c\<^sub>2 \<in> redexes c''"
\<phi>: shows "\<guillemotleft>\<phi> : a \<Rightarrow>\<^sub>B w\<guillemotright> \<and> B.iso \<phi>"
simps_20: fixes sh :: "sheap"   and C :: "char list"   and e :: "'a exp"   and Cs :: "char list list"   and e' :: "'a exp" shows "iconf sh (RI (C,e) ; Cs \<leftarrow> e') = (iconf sh e \<and> \<not> sub_RI e')"
has_field_mono: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and C' :: "char list" assumes "P \<turnstile> C has F:T in D"   and "P \<turnstile> C' \<preceq>\<^sup>* C" shows "P \<turnstile> C' has F:T in D"
lookup_combine_with_key: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t1 :: "('b, 'a) RBT.rbt"   and t2 :: "('b, 'a) RBT.rbt"   and k :: "'b" shows "lookup (combine_with_key f t1 t2) k = combine_options (f k) (lookup t1 k) (lookup t2 k)"
state_prop_def: fixes P :: "'b \<Rightarrow> bool" shows "\<lceil>P\<rceil> = (\<lambda>\<sigma>. P (\<sigma> 0))"
Q: shows "Esigma 12285 = Esigma 14595"
connected_chain_gen: fixes X :: "'a set"   and \<F> :: "'a set set" assumes "X \<in> \<F>"   and "compact X"   and "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"   and "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S" shows "connected (\<Inter> \<F>)"
K_def: shows "K = (if conv_radius f = \<infinity> then norm z + 1 else (norm z + real_of_ereal (conv_radius f)) / 2)"
filtermap_mono: fixes F :: "'a filter"   and F' :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "F \<le> F'" shows "filtermap f F \<le> filtermap f F'"
Domain_empty_iff: fixes r :: "('e \<times> 'f) set" shows "(Domain r = {}) = (r = {})"
orth: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> T" shows "orthogonal a b"
value_inf_zero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = \<infinity>" shows "x = \<zero>"
one_unique: fixes u :: "'a" assumes "u \<in> carrier G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> u \<otimes> x = x" shows "u = \<one>"
strict_iff_order: fixes a :: "'c"   and b :: "'c" shows "(a < b) = (a \<le> b \<and> a \<noteq> b)"
cexp_cong2: fixes p2 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p2 =o r2"   and "Card_order q"   and "Card_order p2" shows "q ^c p2 =o q ^c r2"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Obj (t \<^bold>\<star> u) = False"
power_both_sides: fixes A :: "real"   and B :: "real" assumes "0 \<le> A"   and "0 \<le> B" shows "(A \<le> B) = (A\<^sup>2 \<le> B\<^sup>2)"
Eps_Opt_eq_Some_implies: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "Eps_Opt P = Some x" shows "P x"
simps_1: fixes t1 :: "('b \<times> nat) tree"   and a :: "'b"   and lva :: "nat"   and t2 :: "('b \<times> nat) tree"   and b :: "'b"   and lvb :: "nat"   and t3 :: "('b \<times> nat) tree"   and c :: "'b"   and lvc :: "nat"   and t4 :: "('b \<times> nat) tree" shows "split \<langle>t1, (a, lva), \<langle>t2, (b, lvb), \<langle>t3, (c, lvc), t4\<rangle>\<rangle>\<rangle> = (if lva = lvb \<and> lvb = lvc then \<langle>\<langle>t1, (a, lva), t2\<rangle>, (b, lva + 1), \<langle>t3, (c, lva), t4\<rangle>\<rangle> else \<langle>t1, (a, lva), \<langle>t2, (b, lvb), \<langle>t3, (c, lvc), t4\<rangle>\<rangle>\<rangle>)"
canonical_factorization: fixes t :: "'c local.term" assumes "Arr t" shows "coherent t = (\<lbrace>t\<rbrace> = local.inv \<lbrace>Cod t\<^bold>\<down>\<rbrace> \<cdot> \<lbrace>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<rbrace> \<cdot> \<lbrace>Dom t\<^bold>\<down>\<rbrace>)"
wtSB_wt: fixes \<pi> :: "'b \<Rightarrow> 'fsym trm"   and x :: "'b" assumes "wtSB \<pi>" shows "wt (\<pi> x)"
hyps: assumes "\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"   and "\<forall>i\<le>x. 0 < rec_exec f (xs @ [i])" shows "\<exists>stp>x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
mtx_times_scaleR_commute: fixes A :: "'a sq_mtx"   and c :: "real"   and B :: "'a sq_mtx" shows "A * c *\<^sub>R B = c *\<^sub>R (A * B)"
simps_2: fixes x :: "char list"   and s :: "char list \<Rightarrow> int" shows "aval (V x) s = s x"
ex_un_null: shows "\<exists>!n. \<forall>f. C n f = n \<and> C f n = n"
grd_comp: fixes x :: "'b" shows "[\<cdot> grd x ] * x = x"
A'_def: shows "A' = foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k]"
final_threadI: fixes s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and x :: "'a"   and final :: "'a \<Rightarrow> bool" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "final x"   and "wset s t = None" shows "final_thread.final_thread final s t"
dom_sep: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C (list2FWpolicy p))" shows "x \<in> dom (C (list2FWpolicy (separate p)))"
H: shows "H \<in> subgroups_of_size (p ^ b)"
no_leading_Nil: fixes P :: "'b \<Rightarrow> bool" shows "no_leading P []"
simps_1: fixes n :: "nat"   and m :: "nat"   and l :: "nat list" shows "acklist (n # m # l) = acklist (ack m n # l)"
Next_not_member: fixes S :: "'a set" assumes "finite S" shows "Next S \<notin> S"
sum_image_le: fixes I :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'b) \<le> g (f i)" shows "sum g (f ` I) \<le> sum (g \<circ> f) I"
RN_deriv_finite: fixes N :: "'a measure" assumes "sigma_finite_measure N"   and "absolutely_continuous P N"   and "sets N = sets P" shows "AE x in P. RN_deriv P N x \<noteq> \<infinity>"
rem_implicit_pres_ops_is_standard_operator': fixes vs :: "(char list \<times> nat option \<times> char list list) list"   and I :: "nat list"   and G :: "(nat \<times> nat) list"   and ops :: "(char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi> :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> (vs, I, G, ops)) \<Longrightarrow> is_standard_operator op"   and "\<pi> \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops (vs, I, G, ops)))" shows "is_standard_operator' \<pi>"
assms_2: shows "hb.concurrent (i, Insert e k) (i', Insert e' (Some m))"
homeomorphism_arc: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" obtains h :: "'b \<Rightarrow> real" where "homeomorphism {0..1} (path_image g) g h"
mset_insert: fixes x :: "'b"   and t :: "'b tree" shows "mset_tree (insert x t) = {#x#} + mset_tree t"
delete_edge_preserve_subgraph: fixes H :: "('v, 'w) graph"   and a :: "'v"   and w :: "'w"   and b :: "'v" assumes "subgraph H G" shows "subgraph (delete_edge a w b H) G"
process_mru_new_votes: fixes r :: "nat"   and v_hist :: "nat \<Rightarrow> 'a \<Rightarrow> 'b option"   and v_f :: "'a \<Rightarrow> 'b option" assumes "\<forall>r'\<ge>r. v_hist r' = Map.empty" shows "process_mru (v_hist(r := v_f)) = process_mru v_hist ++ (\<lambda>p. map_option (Pair r) (v_f p))"
accept\<^sub>G\<^sub>R_LTS: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e"   and \<alpha> :: "(('e \<times> 'd \<times> 'e) set \<times> ('e \<times> 'd \<times> 'e) set set) set" assumes "range w \<subseteq> \<Sigma>" shows "accept\<^sub>G\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = accept\<^sub>G\<^sub>R_LTS (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0, q\<^sub>0, \<alpha>) w"
sums_diff: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and g :: "nat \<Rightarrow> 'b"   and b :: "'b" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n - g n) sums (a - b)"
pow_order_eq_1: fixes a :: "'a" assumes "a \<in> carrier G" shows "a [^] order G = \<one>"
ri: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set todo \<union> set old" shows "root_info_cond ri q"
suffix_length_suffix: fixes ps :: "'b list"   and xs :: "'b list"   and qs :: "'b list" assumes "ps \<le>s xs"   and "qs \<le>s xs"   and "\<^bold>|ps\<^bold>| \<le> \<^bold>|qs\<^bold>|" shows "ps \<le>s qs"
carrier_adjunct_ord: fixes a :: "'a" shows "carrier (adjunct_ord D a) = carrier D \<union> {a}"
map_of_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "map_of (update_with_aux v k f ps) = map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)"
1128: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> (y \<squnion> z)) \<sqinter> (x \<squnion> y) = bot"
apply_cases: fixes y :: "'a \<Rightarrow> 'a" assumes "y \<in> {f. bij f \<and> finite {a. f a \<noteq> a}}" obtains x :: "'a perm" where "y = (\<langle>$\<rangle>) x"
dist_triangle_third: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and e :: "real"   and x3 :: "'a"   and x4 :: "'a" assumes "class.metric_space dist uniformity open"   and "dist x1 x2 < e / 3"   and "dist x2 x3 < e / 3"   and "dist x3 x4 < e / 3" shows "dist x1 x4 < e"
assms_1: shows "Bisimulation_Invariant A B sim1 PA PB"
refl_onP_tranclp: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "refl_onP A r" shows "refl_onP A r\<^sup>+\<^sup>+"
lspasl_magicr_der: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h2 :: "'a" assumes "\<not> (A \<longrightarrow>* B) h2" shows "\<exists>h1 h0. (h1,h2\<triangleright>h0) \<and> A h1 \<and> \<not> B h0"
keys_frag_of: fixes a :: "'b" shows "Poly_Mapping.keys (frag_of a) = {a}"
tree_recurse_simps_2: fixes l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "left (tree_recurse l r x) = map_tree l (tree_recurse l r x)"
distinct_7: fixes x21 :: "fmla"   and x22 :: "fmla"   and x3 :: "fmla" shows "x21 OR x22 \<noteq> neg x3"
equals_supr_Above: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> local.Above B"   and "\<And>a'. a' \<in> local.Above B \<Longrightarrow> (a, a') \<in> r" shows "a = supr B"
porder_onE: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "porder_on A r" obtains "refl_onP A r" and "antisymp r" and "transp r"
2: shows "closedin (top_of_set s) (s \<inter> h -` {a..})"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x3 :: "'b" shows "(case prop\<^sub>n(x3) of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f3 x3"
II: fixes \<sigma> :: "'e" assumes "I {} \<sigma>" shows "P \<sigma>"
vangle_def: fixes u :: "'c"   and v :: "'c" shows "vangle u v = (if u = (0::'c) \<or> v = (0::'c) then pi / 2 else arccos (u \<bullet> v / (norm u * norm v)))"
simps_4: fixes h1 :: "'b tree"   and h2 :: "'b tree" shows "cost Merge [h1, h2] = T_merge h1 h2"
termsOfS_preserves_wls: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and s :: "'sort" assumes "wlsPar P"   and "X \<in> termsOfS P s" shows "wls s X"
minimalize_offending_overapprox_subseteq_input: fixes ff :: "('a \<times> 'a) list"   and keeps :: "('a \<times> 'a) list"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "set (local.minimalize_offending_overapprox ff keeps G nP) \<subseteq> set ff \<union> set keeps"
fps_pow_base: fixes f :: "'a fps"   and n :: "nat" shows "fps_nth (f ^ n) (n * subdegree f) = fps_nth f (subdegree f) ^ n"
Collect_minI_ex2_cut: fixes I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "{k \<in> I. P k} \<noteq> {}" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I \<down>< x. \<not> P y)"
subst_lit_id_subst: fixes L :: "'a literal" shows "L \<cdot>l id_subst = L"
eq: shows "stream stern_brocot_recurse = stream (pure Pair \<diamondop> num \<diamondop> den)"
quad_form_bilinear_form: fixes v :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "quad_form v H = bilinear_form v v H"
Alt_eq_map_rexp_iff_1: fixes r :: "'d rexp"   and s :: "'d rexp"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp" shows "(Alt r s = map_rexp f x) = (\<exists>r' s'. x = Alt r' s' \<and> map_rexp f r' = r \<and> map_rexp f s' = s)"
simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.len plus m u v [] = m u v"
ind_0: fixes P :: "nat \<Rightarrow> bool" shows "ind P 0 = (0::'b)"
interpret_floatariths_map: fixes f :: "'a \<Rightarrow> floatarith"   and xs :: "'a list"   and vs :: "real list" shows "interpret_floatariths (map f xs) vs = map (\<lambda>x. interpret_floatarith (f x) vs) xs"
elims_3: fixes x :: "'b match_expr" assumes "\<not> normalized_nnf_match x" obtains m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = MatchAnd m1 m2" and "\<not> (normalized_nnf_match m1 \<and> normalized_nnf_match m2)"   | va :: "'b match_expr" where "x = MatchNot (MatchNot va__)"   | va :: "'b match_expr" and vb :: "'b match_expr" where "x = MatchNot (MatchAnd va vb)"   | "x = MatchNot MatchAny"
cardFinestpart: fixes X :: "'a set" shows "card (finestpart X) = card X"
endofunctor_axioms: shows "endofunctor (\<cdot>\<^sub>S) L"
less_eq_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "(d1 \<le> d2) = (d1 = d2 \<or> d1 = Low)"
lang_PLUS: fixes xs :: "'b rexp list" shows "lang (PLUS xs) = \<Union> (lang ` set xs)"
assms_1: shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
map_pmf_of_set_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (local.pmf_of_set A) = local.pmf_of_set (f ` A)"
nat_filtration_info': fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f \<circ> pseudo_proj_False n = f"
emeasure_T_state_Nil: fixes s :: "'b"   and o\<^sub>0 :: "'a"   and as :: "'b list" shows "emeasure (T (s, o\<^sub>0)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as [] @- \<omega>'} = 1"
domain_very_strict: fixes x :: "'a" shows "(d x = (0::'a)) = (x = (0::'a))"
u': shows "AE x in M. (\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
the_sym_eq_trivial: fixes x :: "'c" shows "(THE y. x = y) = x"
NA_DA_equiv: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "NA.accepts A w = DA.accepts (na2da A) w"
sum_SDG_param_out_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var"   and x :: "nat"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs"   and "V = outs ! x"   and "x < length outs"   and "n = Formal_out (sourcenode a, x)"   and "n' = Actual_out (targetnode a, x)" shows "n s-p:V\<rightarrow>\<^bsub>out\<^esub> n'"
source_size: fixes c :: "'b word \<Rightarrow> 'c" shows "source_size c = LENGTH('b)"
ik\<^sub>s\<^sub>t_is_rcv_set: fixes A :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t A = {t. receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set A}"
vector_minus_component: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(x - y) $h i = x $h i - y $h i"
le_imp_diff_is_add: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "(j - i = k) = (j = k + i)"
delta_def: shows "delta = truncate_up p (zeta - d_min')"
llength_lmap: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "llength (lmap f xs) = llength xs"
eq: shows "(lp p - lp f) \<oplus> lt f = lt p"
simps_1: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and v0 :: "(char list \<times> v) list" assumes "cakeml_static_env static_cenv" shows "cakeml_static_env.is_cupcake_ns static_cenv (Bind v0 []) = list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) v0"
extra_odd_nodes: shows "{v, v'} \<subseteq> odd_nodes_set G"