hcju/isabelleps huggingface dataset:
price_structure_def: fixes pyf :: "'a \<Rightarrow> real"   and T :: "nat"   and \<pi> :: "real"   and pr :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "price_structure pyf T \<pi> pr = ((\<forall>w\<in>space M. pr 0 w = \<pi>) \<and> AEeq M (pr T) pyf \<and> pr T \<in> borel_measurable (F T))"
tp_minus_keys: shows "tp - len \<in> Mapping.keys a2_map"
unfold_triangle: fixes t :: "triangle" obtains a :: "real" and b :: "real" and c :: "real" where "Rep_triangle t = [a, b, c]" and "a \<in> angles" and "b \<in> angles" and "c \<in> angles" and "a + b + c = 180" and "a \<le> b" and "b \<le> c"
SeqSubstFormP_unique: fixes v :: "tm"   and a :: "tm"   and x :: "tm"   and y :: "tm"   and s :: "tm"   and kk :: "tm"   and y' :: "tm"   and s' :: "tm"   and kk' :: "tm" shows "{SeqSubstFormP v a x y s kk, SeqSubstFormP v a x y' s' kk'} \<turnstile> y' EQ y"
v_def: shows "v = vec nc (\<lambda>i. a (col A i))"
assms_2: fixes S1 :: "'a"   and S2 :: "'a"   and b :: "bexp" assumes "S1 \<le> S2" shows "g b S1 \<le> g b S2"
star_of_1_less: fixes y :: "'a" shows "(1 < star_of y) = ((1::'a) < y)"
distinct_13: fixes x21 :: "char list"   and x22 :: "aexp"   and x51 :: "bexp"   and x52 :: "com" shows "x21 ::= x22 \<noteq> WHILE x51 DO x52"
ballpI: fixes A :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. (x, y) \<in> A \<Longrightarrow> P x y" shows "\<forall>(x, y)\<in>A. P x y"
subspace_def: fixes S :: "'b set" shows "subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *\<^sub>R x \<in> S))"
contI2: fixes f :: "'e \<Rightarrow> 'f" assumes "monofun f"   and "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))" shows "cont f"
cofactor_list_syz_def: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "cofactor_list_syz n b = map (\<lambda>i. proj_poly i b) [0..<n]"
assms_1: shows "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"
sat_fv_cong: fixes \<phi> :: "formula"   and v :: "event_data list"   and v' :: "event_data list"   and \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat" assumes "\<forall>x\<in>fv \<phi>. v ! x = v' ! x" shows "sat \<sigma> V v i \<phi> = sat \<sigma> V v' i \<phi>"
o_apply: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and x :: "'g" shows "(f \<circ> g) x = f (g x)"
span_span: fixes A :: "'a set" shows "More_Modules.ideal (More_Modules.ideal A) = More_Modules.ideal A"
Krein_Milman_Minkowski: fixes S :: "'b set" assumes "compact S"   and "convex S" shows "S = convex hull {x. x extreme_point_of S}"
eq_iff: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (c1.le a b \<and> c1.le b a)"
closure_of_minimal: fixes S :: "'a set"   and T :: "'a set"   and X :: "'a topology" assumes "S \<subseteq> T"   and "closedin X T" shows "X closure_of S \<subseteq> T"
base: fixes r :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set"   and S :: "'b sequent"   and \<Gamma> :: "'b form multiset"   and \<Delta> :: "'b form multiset" assumes "r \<in> Ax"   and "Ax \<subseteq> R"   and "snd (extendRule S r) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "lastRule ( \<Gamma> \<Rightarrow>* \<Delta>, 0) r R"
listgcd: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "some_gcd.listgcd xs dvd x"
hfrefD: fixes f :: "'b \<Rightarrow> 'c Heap"   and g :: "'d \<Rightarrow> 'e nres"   and P :: "'d \<Rightarrow> bool"   and RS :: "('d \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('d \<Rightarrow> 'b \<Rightarrow> assn)"   and T :: "'e \<Rightarrow> 'c \<Rightarrow> assn"   and a :: "'d"   and c :: "'b" assumes "(f, g) \<in> [P]\<^sub>a RS \<rightarrow> T"   and "P a" shows "hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)"
sg12: shows "Suc (t + 2 + k) = t + 3 + k"
inf_hidden: fixes A :: "'a set"   and s :: "'a event list"   and P :: "'a process" assumes "\<forall>t. trace_hide t (ev ` A) = trace_hide s (ev ` A) \<longrightarrow> (t, ev ` A) \<notin> F P"   and "s \<in> T P" shows "\<exists>f. isInfHiddenRun f P A \<and> s \<in> range f"
incomp_cancel: fixes p :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "\<not> p \<cdot> u \<bowtie> p \<cdot> v" shows "\<not> u \<bowtie> v"
eq: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> density_states" shows "trace (wlp S Q * \<rho>) = trace (Q * denote S \<rho>) + trace \<rho> - trace (denote S \<rho>)"
simps_15: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b lambda \<Rightarrow> 'b lambda \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "('b \<Rightarrow> 'b lambda \<times> 'a) \<Rightarrow> 'a"   and x3 :: "'b \<Rightarrow> 'b lambda" shows "rec_lambda f1 f2 f3 (Lam x3) = f3 ((\<lambda>lambda. (lambda, rec_lambda f1 f2 f3 lambda)) \<circ> x3)"
closure_closed: fixes S :: "'c set" assumes "closed S" shows "closure S = S"
iso_verts_inv_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'c" assumes "H.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom G)" shows "iso_verts (H.inv_iso hom) (iso_verts hom u) = u"
l2product_lmult: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real"   and c :: "real" assumes "f square_integrable S"   and "g square_integrable S" shows "l2product S (\<lambda>x. c * f x) g = c * l2product S f g"
item_\<beta>_def: fixes x :: "item" shows "item_\<beta> x = drop (item_dot x) (item_rhs x)"
span_add_eq2: fixes y :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and x :: "'b \<Rightarrow>\<^sub>0 'c" assumes "y \<in> phull S" shows "(x + y \<in> phull S) = (x \<in> phull S)"
irreflexive_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "irreflexive A less \<equiv> \<forall>x. x \<in> A \<longrightarrow> \<not> less x x"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
parts_Un: fixes G :: "msg set"   and H :: "msg set" shows "parts (G \<union> H) = parts G \<union> parts H"
cases: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and a1 :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and a2 :: "'e"   and a3 :: "('d \<Rightarrow> real) set"   and a4 :: "'e"   and a5 :: "('d \<Rightarrow> real) set" assumes "AlphaClosure X"   and "AlphaClosure.step_z_alpha X k a1 a2 a3 a4 a5" obtains A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)" and l :: "'e" and Z :: "('d \<Rightarrow> real) set" and l' :: "'e" and Z' :: "('d \<Rightarrow> real) set" where "a1 = A" and "a2 = l" and "a3 = Z" and "a4 = l'" and "a5 = AlphaClosure.cla X k Z'" and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"
factorization_m_def: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "factorization_m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> f =m smult c (\<Prod>\<^sub># fs) \<and> (\<forall>f\<in>#fs. irreducible\<^sub>d_m f \<and> monic (Mp f))"
stopping_time_cong: fixes F :: "'b \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'b"   and S :: "'c \<Rightarrow> 'b" assumes "\<And>t x. x \<in> space (F t) \<Longrightarrow> T x = S x" shows "stopping_time F T = stopping_time F S"
not_exp_le_zero: fixes x :: "real" shows "\<not> exp x \<le> 0"
elims: fixes x :: "('a, 'b) tm"   and y :: "('a, 'b) tm list" assumes "args x = y" obtains uu_ :: "('a, 'b) hd" where "x = Hd uu_" and "y = []"   | s :: "('a, 'b) tm" and t :: "('a, 'b) tm" where "x = App s t" and "y = args s @ [t]"
UINF_false: fixes P :: "'b \<Rightarrow> 'a upred" shows "UINF P (\<lambda>i. false) = false"
assms_3: shows "arg (z * z1) = arg (z * z2)"
UnE: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
s_prod_idl: fixes x :: "'a" shows "1\<^sub>\<sigma> \<cdot> x = x"
convex_rel_interior_finite_inter: fixes I :: "'a set set" assumes "\<forall>S\<in>I. convex S"   and "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}"   and "finite I" shows "rel_interior (\<Inter> I) = \<Inter> {rel_interior S |S. S \<in> I}"
completeness: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c c' Q" shows "ir_hoare P c c' Q"
mult_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c * a \<le> c * b"
minf_5: fixes t :: "'b" shows "\<exists>z. \<forall>x<z. (x < t) = True"
1: shows "FWNormalisationCore.sorted (FWNormalisationCore.insort x xs l) l"
maintain_invar2_ctd_1: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v"   and d :: "nat" assumes "prim_invar2_ctd Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" shows "prim_invar2_ctd (Q' Q \<pi> u) (\<pi>' Q \<pi> u)"
pmf_eq_0_set_pmf: fixes M :: "'d pmf"   and p :: "'d" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
min_eqR: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "min a b = b"
openin_imp_subset: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin (subtopology U S) T" shows "T \<subseteq> S"
dgrad_p_set_subset: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "nat"   and n :: "nat"   and d :: "'d \<Rightarrow> nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "m \<le> n" shows "gd_term.dgrad_p_set pair_of_term d m \<subseteq> gd_term.dgrad_p_set pair_of_term d n"
select_method_wf_mdecl: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and M :: "char list"   and m :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list" assumes "wf_prog wf_md P"   and "P \<turnstile> (C,Cs) selects M = m via Cs'" shows "wf_mdecl wf_md P (last Cs') (M, m)"
FAcc\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls :: "val list"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and v :: "val" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, ls)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, ls)\<rangle>"
app: fixes t :: "'a"   and u :: "'a" shows "(consts (app t u) |\<subseteq>| all_consts) = (consts t |\<subseteq>| all_consts \<and> consts u |\<subseteq>| all_consts)"
mersenne_prime_iff: fixes p :: "nat" shows "mersenne_prime p = (2 < p \<and> prime (2 ^ p - 1))"
integral_diff: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "integrable M f"   and "integrable M g" shows "\<integral> x. f x - g x \<partial>M = integral\<^sup>L M f - integral\<^sup>L M g"
span_eq: fixes S :: "'c set"   and T :: "'c set" shows "(vs2.span S = vs2.span T) = (S \<subseteq> vs2.span T \<and> T \<subseteq> vs2.span S)"
odd_mirror_ap_tree: fixes f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "odd_mirror (f \<diamondop> x) = odd_mirror f \<diamondop> odd_mirror x"
"18_2": fixes x3 :: "(char list \<times> v) list" assumes "allDistinct (pat_bindings p [])"   and "pmatch (c env) (refs st) p v2 [] = Match x3" shows "evaluate_list True (update_v (\<lambda>_. nsAppend (alist_to_ns x3) (sem_env.v env)) env) st [e] (fun_evaluate st (update_v (\<lambda>_. nsAppend (alist_to_ns x3) (sem_env.v env)) env) [e])"
negchecks_model_db_supset: fixes D' :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) set"   and D :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) set"   and F' :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list"   and X :: "'c list"   and \<I> :: "'c \<Rightarrow> ('b, 'c) Term.term"   and F :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list" assumes "D' \<subseteq> D"   and "\<forall>f\<in>set F'. \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> D - D'"   and "negchecks_model \<I> D' X F F'" shows "negchecks_model \<I> D X F F'"
eq: shows "equal_class.equal \<equiv> local.ltl_prop_equiv_quotient_eq_test"
do_if_cases: fixes v :: "v"   and e1 :: "exp0"   and e2 :: "exp0" obtains "do_if v e1 e2 = None"   | "do_if v e1 e2 = Some e1"   | "do_if v e1 e2 = Some e2"
paths_injs_simps: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "(paths t = paths t') = (t = t')"
r'_def: shows "r' = {(x, y). x \<succeq> y \<and> x \<in> c \<and> y \<in> c}"
rref_upt_condition3: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. i < i + (1::'d) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'d)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'d)) $ n \<noteq> (0::'b))"
state_io_rel_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('b, 'c) state" shows "state_io_rel P m = (\<forall>s. P s (snd (run_state m s)))"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "PA a"   and "PB b" shows "sim a b = sim' a b"
finprod_cong': fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finprod G f A = finprod G g B"
bind_refine: fixes M :: "'c nres"   and R' :: "('c \<times> 'd) set"   and M' :: "'d nres"   and f :: "'c \<Rightarrow> 'e nres"   and R :: "('e \<times> 'f) set"   and f' :: "'d \<Rightarrow> 'f nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. (x, x') \<in> R' \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
domain_eq: shows "Domainp (pcr_interval (=)) = (\<lambda>x. x \<in> {(a, b). a \<le> b})"
vec_nth_1_iff_cbox: fixes S :: "('a, 1) vec set"   and a :: "'a"   and b :: "'a" shows "((\<lambda>x. x $ 1) ` S = cbox a b) = (S = cbox (vec a) (vec b))"
mset_replicate_mono: fixes m1 :: "'c multiset"   and m2 :: "'c multiset"   and n :: "nat" assumes "m1 \<subseteq># m2" shows "\<Sum>\<^sub># (mset (replicate n m1)) \<subseteq># \<Sum>\<^sub># (mset (replicate n m2))"
infinite: fixes A :: "'c set" assumes "infinite A" shows "Min A = the None"
ivl_disj_un_singleton_4: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{l<..<u} \<union> {u} = {l<..u}"
i_Exec_Stream_Pre_Post2: fixes c_n :: "'a"   and trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat"   and x_n :: "'b"   and P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "c_n = (i_Exec_Comp_Stream trans_fun input c)\<^bsup>\<leftarrow> c\<^esup> n"   and "x_n = input n" shows "(P c_n \<longrightarrow> Q (input n) (i_Exec_Comp_Stream trans_fun input c n)) = (P c_n \<longrightarrow> Q x_n (trans_fun x_n c_n))"
env_swap: fixes \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list"   and v :: "val"   and e :: "exp" assumes "\<rho> \<approx> \<rho>'"   and "v \<in> E e \<rho>" shows "v \<in> E e \<rho>'"
Par2: fixes Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<longmapsto>\<alpha> \<prec> Q'" shows "P \<parallel> Q \<longmapsto>\<alpha> \<prec> P \<parallel> Q'"
chainI: fixes C :: "'c set set"   and A :: "'c set set" assumes "C \<subseteq> A"   and "\<And>x y. \<lbrakk>x \<in> C; y \<in> C\<rbrakk> \<Longrightarrow> (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x" shows "subset.chain A C"
bindE: fixes p :: "'a + 'c"   and f :: "'c \<Rightarrow> 'a + 'b"   and x :: "'b" assumes "p \<bind> f = Inr x" obtains y :: "'c" where "p = Inr y" and "f y = Inr x"
quat_trad: fixes x :: "real"   and y :: "real"   and z :: "real"   and w :: "real" shows "Quat x y z w = quat_of_real x + \<i> * quat_of_real y + \<j> * quat_of_real z + \<k> * quat_of_real w"
card_of_Field_natLeq_on: fixes n :: "nat" shows "|Field (natLeq_on n)| =o natLeq_on n"
shiftr: shows "Array.length h' a = Array.length h\<^sub>2 a \<and> i' \<le> j \<and> mset ((Array.get h' a)[j := key]) = mset ((Array.get h\<^sub>2 a)[i' := key]) \<and> take j (Array.get h' a) = take i' (Array.get h\<^sub>2 a) @ take (j - i') (drop (Suc i') (Array.get h\<^sub>2 a)) \<and> sorted (take i' (Array.get h\<^sub>2 a) @ take (j - i') (drop (Suc i') (Array.get h\<^sub>2 a))) \<and> (\<forall>x\<in>set (take (j - i') (drop (Suc i') (Array.get h\<^sub>2 a))). key < x)"
rbt_map_entry_rbt_sorted: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "rbt_sorted (rbt_map_entry k f t) = rbt_sorted t"
alpha_2: fixes a :: "vrs"   and x :: "'a"   and b :: "vrs"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
extend_preserves_model2: fixes f :: "nat \<Rightarrow> bool list"   and C :: "fterm literal set" assumes "wf_infpath f"   and "ground\<^sub>l\<^sub>s C"   and "finite C"   and "\<forall>n. \<not> falsifies\<^sub>c (f n) C" shows "eval\<^sub>c HFun (extend f) C"
bind_pause: fixes out :: "'o"   and c :: "'i \<Rightarrow> 'm"   and f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (pause out c) f = pause out (\<lambda>i. bind (c i) f)"
is_shortest_path_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list" shows "is_shortest_path G m n p = (p \<in> path_set G m n \<and> (\<forall>p'\<in>path_set G m n. path_weight G p \<le> path_weight G p'))"
f\<^sub>u: shows "freq (splitLeaf t w\<^sub>a a w\<^sub>b b) = freq u"
update_convs_3: fixes getM' :: "(Literal \<times> bool) list \<Rightarrow> (Literal \<times> bool) list"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getM_update getM' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM' getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
lless_eq: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "partial_order L" shows "(x \<sqsubset>\<^bsub>L\<^esub> y) = (x \<sqsubseteq>\<^bsub>L\<^esub> y \<and> x \<noteq> y)"
s_eq_t: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "\<forall>k\<le>n. \<phi> (the (eval t [f \<triangleright> n])) k = f k"   and "f \<in> U" shows "eval s [f \<triangleright> n] = eval t [f \<triangleright> n]"
connected_open_path_connected: fixes S :: "'c set" assumes "open S"   and "connected S" shows "path_connected S"
mod_eq_nat2E: fixes m :: "nat"   and q :: "nat"   and n :: "nat" assumes "m mod q = n mod q"   and "m \<le> n" obtains s :: "nat" where "n = m + q * s"
no_leading_append: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and ys :: "'f list" shows "no_leading P (xs @ ys) = (no_leading P xs \<and> (xs = [] \<longrightarrow> no_leading P ys))"
wf_Not: fixes x :: "('idx, 'pred, 'act) formula" assumes "weak_formula x" shows "weak_formula (Formula.Not x)"
snd_swap: fixes x :: "'b \<times> 'c" shows "time (prod.swap x) = hamlet x"
list_all2_rev: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
MM1_def: shows "MM1 \<equiv> M \<and> MM1_axioms"
orth_at__ncop2: fixes V :: "'p"   and X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p" assumes "V \<noteq> X"   and "X OrthAt A B C U V" shows "\<not> Coplanar A B C V"
while_def: fixes p :: "'a"   and x :: "'a" shows "while p do x od = (d p \<cdot> x)\<^sup>\<star> \<cdot> ad p"
c_def: shows "c = 2 * y ^ (2 * n + 1)"
comp_preserves_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f"   and "free_monoidal_category.can C g"   and "partial_magma.dom (free_monoidal_category.comp C) f = partial_magma.cod (free_monoidal_category.comp C) g" shows "free_monoidal_category.can C (free_monoidal_category.comp C f g)"
simps_2: fixes i :: "nat" shows "(_Entry_) \<oplus> i = (_Entry_)"
sorted_wrt_imp_sorted_wrt_insort_wrt: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "transp r"   and "\<And>a. r a x \<or> r x a"   and "sorted_wrt r xs" shows "sorted_wrt r (insort_wrt r x xs)"
pivot_funD_3: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "Suc i < nr" shows "f i < f (Suc i) \<or> f (Suc i) = nc"
add_eq_self_zero: fixes m :: "nat"   and n :: "nat" assumes "m + n = m" shows "n = 0"
finsertI2: fixes a :: "'a"   and B :: "'a fset"   and b :: "'a" assumes "a |\<in>| B" shows "a |\<in>| finsert b B"
A_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set A)"
splits_wrtD_4: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" assumes "splits_wrt (ps, qs) T F"   and "hU \<in> set qs" shows "cone hU \<inter> ideal F = {0}"
prems_4: shows "sorted_wrt (\<lambda>x y. y \<le> x) \<beta>s"
msort_set: fixes xs :: "'a list" shows "set (msort xs) = set xs"