hcju/isabelleps huggingface dataset:
run_owner_v_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_owner_v n s))) \<notin> used s"
cross_orthogonal_transformation: fixes f :: "(real, 3) vec \<Rightarrow> (real, 3) vec"   and x :: "(real, 3) vec"   and y :: "(real, 3) vec" assumes "orthogonal_transformation f" shows "f x \<times> f y = det (matrix f) *\<^sub>R f (x \<times> y)"
distinct_19: fixes x21 :: "'d"   and x22 :: "'a \<Rightarrow> 'a"   and x23 :: "('a, 'b, 'c, 'd) seqp"   and x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp" shows "{x21}\<lbrakk>x22\<rbrakk> x23 \<noteq> x31 \<oplus> x32"
ideD_1: fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "ide a" shows "arr a"
pairwise_image: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and s :: "'b set" shows "pairwise r (f ` s) = pairwise (\<lambda>x y. f x \<noteq> f y \<longrightarrow> r (f x) (f y)) s"
run_put_nondet: fixes put :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'c"   and m :: "('b, 'a) nondetT" shows "run_nondet (nondetM_base.put_nondet put s m) = put s (run_nondet m)"
id_i_take_nth_i_drop: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "f = f \<Down> n \<frown> ([f n] \<frown> f) \<Up> Suc n"
nemp: shows "take (\<^bold>|w\<^bold>| - \<^bold>|x\<^bold>|) w \<noteq> \<epsilon>"
AbstFormP_Neg: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm" shows "{AbstFormP v i x x'} \<turnstile> AbstFormP v i (Q_Neg x) (Q_Neg x')"
map_disc_iff: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c literal" shows "is_pos (map_literal f a) = is_pos a"
index_take_if_index: fixes xs :: "'b list"   and x :: "'b"   and n :: "nat" assumes "index xs x \<le> n" shows "index (take n xs) x = index xs x"
bisim1_match_Some_stk_length: fixes P :: "'a J1_prog"   and h :: "'heap"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr"   and pc' :: "nat"   and d :: "nat" assumes "P,E,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)"   and "match_ex_table (compP2 P) (cname_of h a) pc (compxE2 E 0 0) = \<lfloor>(pc', d)\<rfloor>" shows "d \<le> length stk"
simps_5: fixes a :: "'c"   and b :: "'c" shows "dbm_add (Le a) (Lt b) = Lt (a + b)"
topological_tendstoI: fixes "open" :: "'a set \<Rightarrow> bool"   and l :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter" assumes "class.topological_space open"   and "\<And>S. \<lbrakk>open S; l \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> S" shows "filterlim f (topological_space.nhds open l) F"
Lim_component_le: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and net :: "'b filter"   and i :: "'c"   and b :: "real" assumes "(f \<longlongrightarrow> l) net"   and "net \<noteq> bot"   and "\<forall>\<^sub>F x in net. f x \<bullet> i \<le> b" shows "l \<bullet> i \<le> b"
that: assumes "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))"   and "\<And>vv s val. \<lbrakk>v = In2 vv; normal s1\<rbrakk> \<Longrightarrow> dom (locals (snd s)) \<subseteq> dom (locals (snd (snd vv val s)))" shows "thesis"
z_2: shows "z \<in> (interval_bij (a, b) (- 1, 1) \<circ> g) ` {0..1}"
chain_tllist_llist_of_tllist: fixes A :: "('c, 'b) tllist set" assumes "Complete_Partial_Order.chain local.tllist_ord A" shows "Complete_Partial_Order.chain lprefix (llist_of_tllist ` A)"
antisym: fixes a :: "nat"   and b :: "nat" assumes "gcd_nat.greater_eq b a"   and "gcd_nat.greater_eq a b" shows "a = b"
backward: assumes "comm_monoid_mult_class.coprime (real_of_rat_poly p) (real_of_rat_poly q)" shows "comm_monoid_mult_class.coprime p q"
t_2: shows "\<not> A \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile>\<^sub>c t"
bisim1Sync1: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and V :: "nat"   and e2 :: "'addr expr1" assumes "P,e1,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (sync\<^bsub>V\<^esub> (e') e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
assms_3: shows "\<exists>k<0. H2 = cor k *\<^sub>s\<^sub>m H1"
span_post: fixes v :: "'a"   and r :: "'a"   and t :: "'a"   and g :: "'a" assumes "regular v"   and "vector v"   and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"   and "v * - v\<^sup>T \<sqinter> g = bot"   and "t \<le> v * v\<^sup>T \<sqinter> g"   and "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> \<le> r\<^sup>T * t\<^sup>\<star>" shows "v\<^sup>T = r\<^sup>T * g\<^sup>\<star>"
insert_ops_rem_last: fixes xs :: "('b \<times> 'b option) list"   and x :: "'b \<times> 'b option" assumes "insert_ops (xs @ [x])" shows "insert_ops xs"
choose_works: fixes v :: "'a" assumes "v \<in> S" shows "minimal_good_strategy v (choose v)"
onetwo1: shows "set (labels \<sigma>1) \<subseteq> r \<down>s {fst \<beta>_step} \<and> length (labels \<sigma>2) \<le> 1 \<and> set (labels \<sigma>2) \<subseteq> {fst \<alpha>_step} \<and> set (labels \<sigma>3') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step} \<and> set (labels \<gamma>1''') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step} \<and> snd \<beta>_step = fst \<sigma>1 \<and> lst \<sigma>1 = fst \<sigma>2 \<and> lst \<sigma>2 = fst \<sigma>3' \<and> lst \<sigma>3' = lst \<gamma>1''' \<and> fst \<gamma>1''' = fst \<gamma>3"
compl_mono: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c" assumes "boolean_algebra L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> y \<sqsubseteq>\<^bsub>L\<^esub> \<^bold>-\<^bsub>L\<^esub> x"
hcomp_in_vhom': fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and h :: "'a"   and g :: "'a"   and k :: "'a" assumes "B.hseq \<nu> \<mu>"   and "B.dom \<mu> = f"   and "B.dom \<nu> = h"   and "B.cod \<mu> = g"   and "B.cod \<nu> = k"   and "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<rightarrow>\<^sub>B k\<guillemotright>"   and "src\<^sub>B h = trg\<^sub>B f" shows "\<guillemotleft>\<nu> \<star>\<^sub>B \<mu> : h \<star>\<^sub>B f \<rightarrow>\<^sub>B k \<star>\<^sub>B g\<guillemotright>"
finite_measure_mono: fixes A :: "'c set"   and B :: "'c set"   and p :: "'c spmf" assumes "A \<subseteq> B"   and "B \<in> sets (measure_spmf p)" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf p) B"
image_def: fixes f :: "'m \<Rightarrow> 'l"   and A :: "'m set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
ps': shows "\<forall>a\<in>set ps'. case a of (p, t) \<Rightarrow> p < n \<and> t \<in> A"
H_2: shows "isnpolyh (CN c' n' p') n1"
subgraph_def2: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" assumes "graph G\<^sub>1"   and "graph G\<^sub>2" shows "subgraph G\<^sub>1 G\<^sub>2 = (vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2 \<and> edges G\<^sub>1 \<subseteq> edges G\<^sub>2)"
ih3: assumes "\<Gamma> \<turnstile> e[x::=e\<^sub>2'] : T" shows "\<Gamma> \<turnstile> e' : T"
mem_append: fixes l1 :: "'a list"   and l2 :: "'a list"   and m :: "'a" shows "List.member (l1 @ l2) m = (List.member l1 m \<or> List.member l2 m)"
log_altdef: fixes n :: "nat" shows "Discrete.log n = (if n = 0 then 0 else nat \<lfloor>log 2 (real n)\<rfloor>)"
intro\<^sub>3: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma> {c} \<Gamma>'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
\<psi>_in_hom: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "A.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>B G x\<guillemotright>" shows "\<guillemotleft>\<psi> x g : F y \<rightarrow>\<^sub>A x\<guillemotright>"
l8_6: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B C"   and "Bet A C A'" shows "B = C"
A1: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "1 \<le> x"   and "1 \<le> y"   and "1 \<le> z" shows "(x + y) * (y + z) powr \<beta> \<le> (x + y) powr \<beta> * (x + y + z)"
is_aligned_and_2_to_k: fixes n :: "'b word"   and k :: "nat" assumes "n AND 2 ^ k - 1 = 0" shows "is_aligned n k"
cols_def: fixes xss :: "'b list list"   and n :: "nat" shows "cols xss n = (\<forall>xs\<in>set xss. length xs = n)"
in_orthogonal_complement_basis: fixes S :: "'b set"   and B :: "'b set"   and v :: "'b" assumes "subspace S"   and "independent B"   and "B \<subseteq> S"   and "S \<subseteq> span B" shows "(v \<in> orthogonal_complement S) = (\<forall>a\<in>B. orthogonal a v)"
reachable_da: fixes v :: "'a"   and w :: "'a" shows "(v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w) = (v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w)"
set_eqI: fixes A :: "'g set"   and B :: "'g set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
permutes_inj_on: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and A :: "'b set" assumes "f permutes S" shows "inj_on f A"
lin: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "R (h x) (h y) \<or> R (h y) (h x)"
conth: shows "continuous_on ({0..1} \<times> {0..1}) h"
norm_le_rminus: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x \<le> y - n" shows "x \<le> y + 0"
rightUniqueSetCardinality: fixes x :: "'c"   and R :: "('c \<times> 'd) set" assumes "x \<in> Domain R"   and "runiq R" shows "card (R `` {x}) = 1"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>)" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<BB>\<lparr>Dom\<rparr>"
cases: fixes x :: "tm" obtains y :: "name" and z :: "name" and w :: "name" and x :: "tm" where "atom y \<sharp> (z, w, x)" and "atom z \<sharp> (w, x)" and "atom w \<sharp> x" and "xa__ = x"
dvd_mult_div_cancel: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times a (divide b a) = b"
subdef: shows "usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta> \<noteq> undeft"
AbsCheck: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>\<^bold>\<lambda>x. \<^bold>\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>,x\<rparr> in v] = (proper x \<and> (case rep x of \<omega>\<nu> \<omega> \<Rightarrow> False | \<alpha>\<nu> y \<Rightarrow> True))"
fps_hensel_def: fixes F :: "'b fps poly"   and g :: "'b poly"   and h :: "'b poly" shows "fps_hensel F g h \<equiv> (monic F \<and> reduce_fps_poly F = g * h) \<and> coprime g h \<and> 0 < degree g \<and> 0 < degree h"
Graph_axioms: shows "Graph (ls.\<alpha> \<circ> succ)"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (src f)"
simps_6: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Arctan a) vs = arctan (interpret_floatarith a vs)"
chain_sup_const: fixes S :: "('e, 'f) labeled_graph" shows "chain_sup (\<lambda>x. S) = S"
cones_map_ide: fixes \<chi> :: "'a \<Rightarrow> 'b"   and a :: "'b" assumes "\<chi> \<in> D'.cones a" shows "D'.cones_map a \<chi> = \<chi>"
power_is_relpow: fixes X :: "('a \<times> 'a) set"   and n :: "nat" shows "rel_dioid.power X n = X ^^ n"
comp_ipresIGWlsAll: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('o, 'p, 'q, 'r, 's, 't, 'k, 'm, 'u) model_scheme"   and MOD' :: "('v, 'w, 'q, 'r, 'x, 'y, 'l, 'n, 'z) model_scheme"   and h' :: "'l \<Rightarrow> 'aa"   and hA' :: "'n \<Rightarrow> 'ab"   and MOD'' :: "('ac, 'ad, 'q, 'r, 'ae, 'af, 'aa, 'ab, 'ag) model_scheme" assumes "ipresIGWlsAll h hA MOD MOD'"   and "ipresIGWlsAll h' hA' MOD' MOD''" shows "ipresIGWlsAll (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
id: shows "det (four_block_mat A1 A2 B3 B4) = det A1 * det B4"
in_smallomega_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "f \<in> \<omega>[F](\<lambda>x. 0::'b)"
rep_eq: fixes x :: "complex" shows "Rep_moebius_mat (blaschke_mmat x) = blaschke_cmat x"
waiting_def: fixes w :: "'a wait_set_status option" shows "waiting w = (\<exists>w'. w = \<lfloor>InWS w'\<rfloor>)"
r_singleton_encode_def: shows "r_singleton_encode \<equiv> Cn 1 S [Cn 1 r_prod_encode [Z, recf.Id 1 0]]"
assms: shows "as \<in> carrier (RDirProd_list Rs)"
lfp_funpow: fixes f :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "mono f" shows "lfp (f ^^ Suc n) = lfp f"
case_distrib: fixes h :: "'e \<Rightarrow> 'd"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and option :: "'f option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
cases: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "('c, 'd) label"   and a3 :: "'c \<Rightarrow> 'd" assumes "ce a1 a2 a3" obtains \<sigma> :: "'c \<Rightarrow> 'd" where "a1 = \<sigma>'''__" and "a2 = Skip" and "a3 = \<sigma>'''__"   | e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" and \<sigma> :: "'c \<Rightarrow> 'd" where "a1 = \<sigma>''''__" and "a2 = Assume e__" and "a3 = \<sigma>''''__" and "e__ \<sigma>''''__"   | \<sigma> :: "'c \<Rightarrow> 'd" and v :: "'c" and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" where "a1 = \<sigma>" and "a2 = Assign v e" and "a3 = \<sigma>(v := e \<sigma>)"
1: shows "\<guillemotleft>f \<star> \<iota> : f \<star> a \<star> a \<Rightarrow>\<^sub>S f \<star> a\<guillemotright>"
assms_1: shows "eq_\<I>_gpv A \<I> gpv1 gpv1'"
assms_1: fixes n :: "nat" shows "l \<le> lower_asymptotic_density (A n)"
simps_22: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b" shows "(case Void of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f1"
rel_fun_conversep: fixes A :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "A\<inverse>\<inverse> ===> B\<inverse>\<inverse> = (A ===> B)\<inverse>\<inverse>"
constrains_Int: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A co A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B co A' \<inter> B'"
Start: fixes b :: "(nat \<times> int) list"   and ds :: "nat set" shows "b \<in> grid b ds"
Compl_greaterThan: fixes k :: "'c" shows "- {k<..} = {..k}"
compE_def: fixes \<rho> :: "'a \<Rightarrow> 'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and x :: "'a" shows "I.compE \<rho> \<xi> x \<equiv> I.int \<xi> (\<rho> x)"
l_repr_independence: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <#\<^bsub>add_monoid G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x <#\<^bsub>add_monoid G\<^esub> H = y <#\<^bsub>add_monoid G\<^esub> H"
BDij: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "(B * D) $$ (i, j) = B $$ (i, i) * D $$ (i, j)"
I_hd_coeff1_mult: fixes m :: "int"   and \<phi> :: "atom fm"   and x :: "int"   and xs :: "int list" assumes "0 < m"   and "qfree \<phi>"   and "\<forall>a\<in>set (atoms\<^sub>0 \<phi>). hd_coeff a dvd m" shows "Z.I (map\<^sub>f\<^sub>m (hd_coeff1 m) \<phi>) (m * x # xs) = Z.I \<phi> (x # xs)"
domain_def: fixes none :: "'b"   and f :: "'a \<Rightarrow> 'b" shows "opt.domain none f \<equiv> {x. f x \<noteq> none}"
ifTrue: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e1 :: "('r, 'l, 'v) expr"   and e2 :: "('r, 'l, 'v) expr" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1])))"
ck_unifier_def: fixes t :: "'b trm"   and s :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and type :: "inferences" shows "ck_unifier t s \<sigma> type = (if type = FirstOrder then MGU \<sigma> t s else Unifier \<sigma> t s)"
begin_correct: fixes n :: "nat" assumes "0 < n" shows "{inv_begin1 n} tcopy_begin {inv_begin0 n}"
invariant_1D_2: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "sgn (rai_lb plr) = sgn (rai_ub plr)"
assms_2: fixes x :: "constraint" assumes "x \<in> set cs" shows "\<nexists>a b. GT a b = x"
iface_subset_def: fixes i1 :: "iface"   and i2 :: "iface" shows "iface_subset i1 i2 = internal_iface_name_subset (iface_sel i1) (iface_sel i2)"
simps_2: fixes h :: "'heap" shows "typeof\<^bsub>h\<^esub> Unit = typeof Unit"
\<Psi>_def: shows "\<Psi> = \<lparr>edge = edge', capacity = cap, source = None, sink = \<langle>a\<rangle>\<rparr>"
yx_3: shows "y \<notin> Fvars \<psi>"
homeomorphic_map_compactness: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f"   and "U \<subseteq> topspace X" shows "compactin Y (f ` U) = compactin X U"
after_summary_zmset_of_nonneg: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'c multiset"   and S :: "'b antichain"   and t :: "'c" assumes "dataflow_topology summary results_in" shows "0 \<le> zcount (dataflow_topology.after_summary results_in (zmset_of M) S) t"
simps_2: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "rn_rev_price N der matur (Suc n) w = discount_factor r (Suc 0) w * expl_cond_expect N (proj_stoch_proc geom_proc (matur - Suc n)) (rn_rev_price N der matur n) w"
add: shows "basis_reduction_add_rows_cost upw i state = (state', c_add)"
get_element_by_is_l_get_element_by: shows "l_get_element_by Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name Shadow_DOM.to_tree_order"
F_ndet: fixes P :: "'b process"   and Q :: "'b process" shows "F (P |-| Q) = F P \<union> F Q"
\<tau>move1SeqRed: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and e :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (Val v;; e)"
scaleC_scaleC: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "complex"   and b :: "complex"   and x :: "'a" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (scaleC b x) = scaleC (a * b) x"
sound_mode_use\<^sub>A: shows "abs.sound_mode_use gc\<^sub>A"
sorted_dropWhile_filter: fixes xs :: "nat list"   and I :: "\<I>"   and nt :: "nat" assumes "sorted xs" shows "dropWhile (\<lambda>t. enat t + right I < enat nt) xs = filter (\<lambda>t. \<not> enat t + right I < enat nt) xs"
elims_2: fixes x :: "'b literal" assumes "positive_literal x" obtains xa :: "'b equation" where "x = Pos xa"
ldropn_iterates: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "ldropn n (iterates f x) = iterates f ((f ^^ n) x)"
c6: assumes "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "P (rt(ip \<mapsto> addpre (the (rt ip)) (\<pi>\<^sub>7 r)))"
l: shows "Vagree \<nu> (stateinterpol \<nu> \<nu>' V) V"
2: fixes k :: "nat"   and x :: "'a" shows "indicat_real (\<Union> (S ` {..k})) x \<le> indicat_real (\<Union> (S ` {..Suc k})) x"
split0: fixes t :: "tm"   and vs :: "'b list"   and bs :: "'b list" shows "tmbound 0 (snd (split0 t)) \<and> Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t"
not_empty: shows "space (point_measure keys (\<lambda>x. ennreal (K x))) \<noteq> {}"
scalar_product_CONS: fixes xs :: "nat list"   and bs :: "bool list"   and "is" :: "int list" assumes "length xs = length bs" shows "scalar_product (map_index (\<lambda>i n. 2 * n + map of_bool bs ! i) xs) is = scalar_product (map of_bool bs) is + 2 * scalar_product xs is"
base: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<phi> xs" shows "ev \<phi> xs"
sel_1: fixes r :: "real"   and x :: "complex" shows "Re (r *\<^sub>R x) = r * Re x"
assms_1: shows "proj \<in> quasi_measure_preserving M M2"
shadows_consts_sterm_simps_3: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and cs :: "(Term_Class.term \<times> sterm) list" shows "pre_constants.shadows_consts C_info heads (Sabs cs) = list_ex (\<lambda>(pat, t). \<not> fdisjnt (pre_constants.all_consts C_info heads) (frees pat) \<or> pre_constants.shadows_consts C_info heads t) cs"
Min_gr_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Min less_eq A) = (\<forall>a\<in>A. less x a)"
elims_3: fixes x :: "('c, 'd) ioa"   and xa :: "'c \<times> ('d \<times> 'c) list" assumes "\<not> is_exec_frag_of x xa" obtains A :: "('c, 'd) ioa" and s :: "'c" and ps :: "('d \<times> 'c) list" and p' :: "'d \<times> 'c" and p :: "'d \<times> 'c" where "x = A__" and "xa = (s__, Cons p__ (Cons p' ps))" and "\<not> (snd p' \<midarrow>fst p__\<midarrow>A__\<longrightarrow> snd p__ \<and> is_exec_frag_of A__ (s__, Cons p' ps))"   | A :: "('c, 'd) ioa" and s :: "'c" and p :: "'d \<times> 'c" where "x = A" and "xa = (s, [p])" and "\<not> s \<midarrow>fst p\<midarrow>A\<longrightarrow> snd p"
n_less_Suc: fixes n :: "nat" shows "n < Suc n"
dom_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "dual_category C" shows "partial_magma.dom (dual_category.comp C) f = partial_magma.cod C f"
alternating_harmonic_series_sums: shows "(\<lambda>k. (- 1) ^ k / real (Suc k)) sums ln 2"
pdivides_imp_root_sharing: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list"   and a :: "'c" assumes "domain R"   and "p \<in> carrier (poly_ring R)"   and "p pdivides\<^bsub>R\<^esub> q"   and "a \<in> carrier R"   and "ring.eval R p a = \<zero>\<^bsub>R\<^esub>" shows "ring.eval R q a = \<zero>\<^bsub>R\<^esub>"
Step_Lrule: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and L :: "lrule"   and j :: "nat" shows "step_result (SG, C) (i, Lrule L j) = (close (SG @ Lrule_result L j (SG ! i)) (SG ! i), C)"
cases: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" assumes "rightPrincipal a1 a2" obtains C :: "'a sequent" and F :: "'a" and Fs :: "'a form list" and Ps :: "'a sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<Empt> \<Rightarrow>* Compound F Fs)"
assms_3: shows "(V, as'', as') \<in> dependent_live_vars n'"